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This project is analysis of the seismic data by some of the existing procedures used in the correction of seismic data .The project traces the development of the correction procedures and describes their rationale and methodology. And also proposes modifications to the existing procedures and provides software listing used in this study as a step in the interests of good practice in disseminating information on seismic correction. Moreover it proposes several metrics for judging the reliability of corrected data through the use of power spectral densities, phase spectra, coherence estimates, acceleration response spectra and the short-time Fourier transform and draws conclusions on the reliability of some of the correction procedures used. The Least Mean Squares (LMS) algorithm and the square root, Recursive Least Squares (RLS) algorithms are considered and investigated using three seismic events. Both adaptive methods do not assume any knowledge of instrument data, but use seismic readouts from which to estimate the inverse instrument response. The project shows that in the absence of instrument data, adaptive methods provide reasonably consistent acceleration response spectra and power spectral densities. In this project I used QR-RLS algorithm, my main objective of the project is, by understanding the various earthquake incidents in detail and to find facts, recover the actual data by using various seismic methodologies and represent them in logical graphical format.
Earthquakes are the most devastating natural events that occur on earth. The Study of these Earthquakes is a part of Seismology. Seismology is the scientific study of earthquakes and th propagation of elastic waves through the Earth. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes. Most earthquakes occur at depths of less than 80 km (50 miles) from the Earth's surface When the Chilean earthquake occurred in 1960, seismographs recorded seismic waves that traveled all around the Earth. These seismic waves shook the entire earth for many days! This phenomenon is called the free oscillation of the Earth In earthquake engineering analysis and particularly the dynamic behavior of structures, the importance of credible ground motion time series cannot be underestimated. Reliable and extensive sets of ground motion time-series, recorded from actual earthquakes, are essential. In most cases however seismic data sets have insufficient information regarding the type of recording instrument used, furthermore in a lot of cases information on the instrument is simply not available and researchers clearly state that instrument correction is not applied to the data. In earthquake engineering analysis and particularly the dynamic behavior of structures, the importance of credible ground motion time series cannot be underestimated. Reliable and extensive sets of ground motion time-series, recorded from actual earthquakes, are essential. In most cases however seismic data sets have insufficient information regarding the type of recording instrument used, furthermore in a lot of cases information on the instrument is simply not available and researchers clearly state that instrument correction is not applied to the data.
Forecasting a probable timing, location, magnitude and other important features of a forthcoming seismic event is called earthquake prediction. Most seismologists do not believe that a system to provide timely warnings for individual earthquakes has yet been developed, and many believe that such a system would be unlikely to give significant warning of impending seismic events. More general forecasts, however, are routinely used to establish seismic hazard. Acceleration time-series are records the seismic data over the entire duration of Earth quake. In this actual ground motion is convolved with the instrument response. The main aim of the project is to recover the real actual data by using various seismic methodologies.
The sources used for this application are:
Operating System : Windows XP SP2 or Linux
Mathematical Computing : Matlab V6.5 R13
Office Product Suite : MS Office 2007
Processor : Pentium III or Equivalent
Clock Speed : 233 MHz
Monitor : Standard VGA Display
RAM : 128MB
Hard Disk : 2 GB
Key Board : 101 Key Based
Network Interface Card : Any
The research is mainly about the analysis of seismic data by various correction techniques, various logarithm. Those are the explained below are:
Novel Seismic Correction approaches without instrument data, using adaptive methods and De-Noising:
This provides the comparison between the two techniques of de-convolving the instrument reply from seismic incident against single degree of freedom technique. The Recursive Least Squares algorithm, square root algorithm and Least Mean Squares algorithms are analyses by three seismic incidents. Any knowledge of instrument information do no assume by both the techniques. This paper demonstrates that in the instrument information unavailability, adaptive techniques give reasonable reply spectra as well as power spectral densities. In the analysis of earthquake engineering, we cannot underestimate the significance of ground motion time series. Almost in all the cases, the seismic information sets contains insufficient data about the instrument used for recording. Some of the corrections methods are used for digitizing the information, instrument correction and de noise by the wavelet transform. In the signal processing point of view, a seismic instrument for recording should provide the response. Most of the seismic information even considers a second order, single degree of freedom instrument task with which to de convolve the reply of instrument form the motion of ground. This paper provides the comparison between the QR-recursive least squares (RLS) algorithm and the Least Mean Squares algorithm's inverse filtering implementation. For de convolving the instrument reply, the resulting inverse filters are implemented to the information. This method does not need any details about the instrument. It needs only the data, which is provided by the instrument to find the inverse of instrument response. This is advantage of this technique. The implementation of the translation invariant wavelet transform is discussed in this paper.
Most seismic correction methods apply a 2nd order, single-degree-of-freedom (SDOF) instrument function with which to inverse filter or de-convolve the accelerometer response. To obtain estimates of the ground acceleration from the recorded relative displacement response, the SDOF instrument correction is applied as follows:
The above expression is used to de-convolve the accelerometer response
A Review of Procedures used for the Correction of Seismic data:
This report provides few of the previous functionalities applied in the correction of seismic Information. It concludes that a common requirement of data on the correction methods applied, with important exceptions, when providing CD-corrected information makes it hard to describe conclusions on the consistency of the proper information records.
Essential information for earthquake technology is found from evaluates from earth quaking during earth shake. The first accurate measurements of destructive earthquake ground motions were made during the Long Beach, California earthquake of 10/03/33 (Hudson ). Analogue techniques are the easiest elements that are sensibly economical to construct and need minimum maintaining cost. However information from these tools needs wide data-processing clock time. Analyses handled by Shakal . advise that digital Tools on the different hand, although more costlier to maintain, allow a more correct determination of earth move and decrease data-processing clock time. The first ever attempt to devise a procedure to correct recorded accelerograms was made in the 1970's by Trifunac et al.  In correct registered accelerograms process, the new information is initial low-pass filtered to ignore high bandwidth interference. The information is then tool is proper followed by high-pass filtering to ignore baseline error. This method makes utilize of an Ormsby filter. This program adopts a correction process, which contains adaptive filtering possibly in place of tool correction, however the information are to scant to be of any application.
Table 1 below summarizes some of the methods used for correcting seismic data.
Due the difficulty of using BAP the correction procedure discussed here has been coded entirely in Matlab [a] , see appendix. It follows many of the features of BAP.
Elements of Correction Procedures
1. Interpolating, Re-sampling
The first step in all correction procedures requires interpolation such that the points are equally spaced. In the strong motion data
examined in this study a sampling rate of 600 Hz is used In correction modules (BAPS) information was divided and recombined by a standard overlapped technique. The proposed UEL correction technique bypasses segmentation and its related operations.
2. Baseline Error correction, De-trending
Velocity and displacement calculated from such an accelerograms will resultant in a linear and quadratic equation fault respectively. The baseline correction [Matlab:detrend] is executed by reducing the least-square regression line functions from the accelerograms.
3. Instrument Response
Instrument correction is essential to find a better calculation of the earth movement. The tool itself represents molded as SDOF structure and its dynamic attributes evaluated.
The pattern is then applied to decouple the tool response from the current earth movement.
The equation of motion of the model mass given by;
where Ï‰i is the natural angular frequency and Î³ is the ratio of critical damping of the instrument. By using the Fourier transform, the equation (1) can be transformed into equation (2). Where the approximate acceleration output of the instrument A(f ) X (f ) i=Ï‰ 2 .
Thus the Fourier transform of the ground acceleration X ( f ) g can be recovered from knowledge of the relative displacement of the instrument X ( f ) . The complex transfer function H(f) is multiplied by the approximated acceleration frequency content A(f).That the acceleration is approximately equal to the ground acceleration for frequencies of up to about 10 Hz. Trifunac's method (NOAA) performs decimation prior to instrument correction, it is there fore possible that the movement of the ground and instrument itself are only partially de-coupled.
4. Filtering and Phase Correction
It would be important if calculates of local interference indications are added in any strong motion information as an indicantion of the local signal to noise ratio.
This is fundamental since phase content inbuilt in a seismic trace checks the occurrence of peaks and should therefore be maintained without any unnecessary distortion.
5. Decimation, Down Sampling
Decimation is not needed for information already in digital type from SSA-1 at 200Hz. Decimation requires the refusal of those data which lie with in the necessary clock time interval.
6. Adaptive methods
In specific we assume a least-squares adaptive method. The two most broadly applied adaptive methods are those which minimize the mean-square computer error (Least Mean Squares) and the recursive least-square computer error.
The following instruments are applied in the numerical analyze to evaluate the efficiency of correction processes.
1. Fourier Spectra
2. Ground Motions and Power Spectral Estimates
3. Time-frequency distributions, Spectrograms
4. Coherency function 5. Linear Total Acceleration Response Spectrum
1. Nahanni Aftershock, Battlement Creek, 360, USA
It is helpful to mark that the coherence is successfully a relative comparability of the similarity of bandwidth components but it doesn't suggest by itself the magnitude and phase angle of these components. Hence it requires being access with the power and phase angle spectra simultaneously
The acceleration response spectrum for the Nahanni after-shock demonstrates that selecting an unsuitably low rate for the higher cut bandwidth can acquire a computer error of the order of 20% in the response spectrum. This is minimally too long to ignore from a structural point of view. The instrument correction is indicated to change the phase angle report of the corrected register. Also the requirement for ero-phase filtering at entire level of the correction is underlined. Additional exercise need to clear up the effects of adaptive filtering on seismic data.
A New Approach to Seismic Correction using Recursive Least Squares and Wavelet De-Noising:
This report talks about a relatively direct execution of the known RLS (Recursive Least Squares) algorithmic program in the situation of a structure recognition problem. The resulting inverse filter is so used to the information in order to de-convolve the tool reply. The report matches power spectral plots and the overall acceleration response spectra of two pair of seismic cases.
Displacement based structure and performance based structure techniques are becoming more popular and executable. However the tools that register the earth acceleration are not proper, and commonly register a clock time series which requires to be modified to regain the "original earth move" itself. A correction method requires to
digitalize, that is equivalent sample the data,
correct for instrument functionalities
(iv) de-noise, by band pass filtering or wavelets,
(v) resample to a suitable sampling rate.
For some databases , the whole issue of correcting for an unknown instrument is too problematic hence no instrument deconvolution is performed. The authors did not want to present time series ground motion that have imposed and incorrect, corrections. However they present data with no instrument correction which is hardly what an engineer would usefully want.The Recursive Least Squares algorithmic program is applied to check, post-priori, the filter feature or fingerprint, if you like, that the tool allows enforced on the clock time series.
Inverse filtering by the Recursive Least Squares algorithmic program
The Recursive Least Squares algorithmic program [4,5,6,7] was selected in priority to the LMS (Least Mean Squares) adaptive algorithmic program. One cause is that the Recursive Least Squares algorithmic program is depending on the arriving information samplings rather than the statistics of the ensemble common as in the instance of the Least Mean Squares algorithmic program. The Recursive Least Squares is comparatively direct to utilize because ultimately only the forgetting element of requirements to be adapted. This is utilized to decrease the rate of older error information.
QR-RLS inverse-filter frequency responses with wavelet de-noising:
The frequency-response characteristics were obtained after the information was wavelet de-noised. The tool attributes for these cases are 10Hz for the instrument time and 0.552 damping. Noise computer error must, as far as is possible, be separate before a Recursive Least Squares tool correction is utilized, since the resulting de-convolution may amplify the interference inherent during the seismic information collection and distort the bandwidth response.
The report shows that inverse filtering applying the Recursive Least Squares algorithmic program returns acceptable final results when evaluate to the common second order type de-convolution. This is matched to applying a second order differential solution in either the time period or frequency region.
In most instances however seismic information sets have lacking data about the type of registering tool utilized, furthermore in several types data on the tool is not accessible and researchers correctly submit that instrument correction isn't utilized to the information. The Recursive Least Squares algorithmic program however offers a result to the above problem. It is best suggestion of the actual instrument reply. The sequence of actions is also significant since the information must be de-noised or filtered before the tool adjustment. This is to avoid any intensification of disturbance in the de-convolution procedure. The outcomes however do demonstrate that band widths of interest it is still probable to get more or less zero magnitude and phase reaction while de-convolving before de-noising or filtering, however it is also clear that at higher bandwidths distortion became clearer.
De-Convolution Of Seismic Data Using Partial Total Least Squares:
The Partial Total Least Squares (PTLS) [Demmel 1987, Golub and Van Loan 1996, Golub, Hoffman and Stewart 1987](8) is a difference on the TLS (Total least Squares) [Golub and Van Loan 1996](9) which considers that whole the measurents on both faces of Ax = b are noisy. The seismic information applied for the study is selected as much as possible specified tool attributes were retained in the header documents of the seismic register [Ambraseys et al, 2000, Converse, 1992].(10)
Seismic correction techniques commonly implement a second order, SDOF (single-degree-of-freedom) instrument application with which to inverse filter or de-convolute the accelerometer reply. Therefore to get approximations of the base acceleration from the registered reply, the single-degree-of-freedom instrument correction is utilized as follows:
where ïƒ£ï€ is the accelerometer viscous damping ratio, ïƒ¹ï€ is accelerometer natural frequency and a g (t ) is the
QR-Recursive Least Squares (QR-RLS) method
QR-decomposition related Recursive Least Squares algorithmic program is calculated from the square-root Kalman filter counterpart [Haykin 1996](11), [Sayed and Kaileth 1994](12). The 'square-root' is in fact a Cholesky factorization of the inverse correlation matrix. The source of this algorithmic program depends upon the function of an orthogonal triangulation module known as QR decomposition.
The Total Least Squares (TLS):
The Total Least Squares [Golub and Reinsch 1970, Golub and Van Loan 1996, Van Huffel and Vanderwalle 1985, 1988, 1991, 1980] (8)contains some programs in de-convolution in medicine and spectroscopy. It was implemented to the de-convolution [Chanerley and Alexander 2006](12) of seismic information in order to find an estimation of the device reply. This technique of de-convolution has the benefit which it contains the computer error in the sensibility matrix as well as the information vector. Generally the Total least Squares algorithmic program established that it could be applied successfully to de-convolute the device outcome from the seismic information [Chanerley and Alexander 2006](13). However, computationally the Total least Squares algorithmic program needs a large value of computer memory when processing with wide information sets and in repeat precision.
The seismic data applied for the outcomes found is from the SMART-1 [Abrahamson et al 1987] (14)array which has the detectors placed down in concentric rings and is placed in the north-east corner of Taiwan nearly the city of Lotung. In the study of the SMART-1 array information there is not any requirement for very much filtering except at the less bandwidths and baseline rectification, since the low-pass Butterworth filter contains frequency fixed the signal. It has been represented that the PTLS algorithmic program gets fine outcomes where the unmodified rows and columns are [1 x 1] when evaluate to the default Total least Squares. With several rows and columns unmodified the inverse responses are not representative of the device outcome. So there is not a net improvement in utilizing the PTLS as against to the Total least Squares. Almost the entire test shown in this report further low-pass filtering was avoided. Wavelet de-composition and Reconstruction was used in order to remove certain low bandwidth and high bandwidth item after de-convolution.
Lp Deconvolution of Seismic Data Using the Iterative Re-Weighted Least Squares Method:
In This technique of de-convolution was applied on synthetic and real seismic information [15,16] in order to find its sensitivity to explode of noise. In this report the Iterative Re- Weighted Least Squares is used as a common de-convolution method for de-coupling the device outcome from the seismic information in order to find an estimation of the true earth movement. The Iterative Re-Weighted Least Squares was utilized to the seismic information later on de-noising and correcting for baseline, but without any frequency selective filtering.
Although Lp optimizations was used to information, which was considered to have interference bursts affected in that [15,16], it has been discovered that it allows a Sensible technique of de-convoluting the device outcome from the seismic information. This SMART-1 [18, 19] array contains the detector arranged in concentric rounds and is placed in the north-east angle of Taiwan near the city of Lotung. A specified problem in seismic correction technologies is that rather often the transfer procedure of the recording device is not recognized; in especial in certain older (legacy) data. Where device attributes are allowed, a second order single-degree-of-freedom transfer procedure is used in either the time or frequency domain  in order to decouple the tool reply. In this report the iterative re-weighted least squares (IRLS) Lp optimization is used which relies on a weighted least square solution, applying different weighting for every procedure.
In considering the information to be examined, in specific, the SMART-1 information, then it is Significant that de-convolution is applied before any upcoming processing of the recorded information if filtering rather than de-noising is implemented. If this sequence is not observed then of course the Iterative Re-Weighted Least Squares algorithmic program (or some of the algorithmic programs applied for de-convolution) will de-convolute the utilized filter simultaneously with the carry-over application(s) inherent in the information due to the tool. The paper indicate that Lp de-convolution gives a full inverse calculate of the real Butterworth filter used and therefore is a reasonable method to apply for de-convolution where instrument attributes are not available.
In conclusion the Lp de-convolution technique used to seismic information gives a better computation of the inverse of the instrument features and can be applied to calculate device outcome where none is accessible. When applying wavelet de-noising the order is not critical since the approach applies perfect reconstruction filters after the function of a threshold. If this is not followed then the de-convolution method will simply acquire and calculate of the last filter applied. Application of the wavelet transform to seismic information at various levels establishing either the low bandwidth or high-frequency detail is a valuable addition to the analyzing instruments usable for seismic verification.
Modeling Non-Linear Effects in Seismic Data from Estimates of Bispectra Using Linear Prediction and Volterra Kernels:
This report continues from latest process on the adjustment of seismic information using structure identification techniques, which calculate the bandwidth responses of an accelerometer in order to reverse engineer and get a better computation the earth movement time. In order to calculate the bispectra of seismic time-histories using a non-linear pattern, the attributes of the linear part of the total pattern are first calculated by a linear predictive coding method. The linear pattern is then translated to the time domain and applied to extract the linear element from the information.
Recent investigations on the correction of seismic data , which developed on previous function would suggest that very few of the frequencies in the acceleration response spectra and might be artifacts and must be unobserved. This report applies a common, linear forecasting technique (Levinson-Durbin) in order to calculate pattern coefficients. The report also analyzes the possible presence of non-linear effects in the seismic signal, by fitting a second order Volterra pattern to the residue signal.
In calculating the coefficients of the linear predictive pattern, the chosen information is first coordinated into overlapping sections. The quantity of overlap and section length may be changed and were indeed acquired to their extreme points, but found to be uniformly stabile with only small dissimilarities. Coefficients of the linear predictive pattern are calculated for every section and the power spectrum of the section is calculated applying the predictive pattern. In order to calculate the coefficient vector of the overall linear predictive pattern the averaged out value of the coefficient vectors of entire sections is calculated. However the importance of the calculated time-history in this study lies in the fact that it is a first-order-effect time-history. This is subtracted from the registered seismic case (i.e. the observed output).
In order to calculate the transform of the second-order kernel of the Volterra pattern we form the input-output across bi-correlation application. This review indicates that bandwidth distortion may happen and can show up in the acceleration response spectra. It shows the importance of extending the analysis of seismic events applying the bispectrum and a non-linear Volterra pattern. This study can be reached higher order Volterra patterns. The source of such as artifacts could be the seismic detector or some of the filtering levels, based on the type of filter applied. Indeed they may be a specified key signature of a devices based. On the a different reason could be entirely external to the accelerometer, either from the system in which the speedometer is enclosed or indeed from the earth movement itself, in which case it may not be an artifact. Nevertheless the outcomes represent that bandwidth coupling does happen and can be found in seismic events and give rise to bandwidth artifacts.
An Approach to Seismic Correction which includes Wavelet De-noising:
This report starts with a short introduction to few techniques applied to adjust seismic information . It defines common techniques of de- convolving devices and functional outcome from seismic accelerograms. These are the convolution of earth movement with the carry-over application of the recording devices and system on which the devices is placed.
Several example seismic signals are threshold de-noised using the stationary wavelet transform (SWT) and compared with the more common band-pass filtering methods. The report comparisons between power spectral plots and the total acceleration response spectra of earth shake applying the band-pass filtering and wavelet de-noising techniques.
Converse (BAP) interpolates to 600Hz, then implements a baseline correction, segments and zero-pads the information and implements a cosine taper window.
UEL (A) is similar to BAP except that zero-phase filtering is applied in every instances; the Information is processed end-to-end as one section without taking to apply a cosine taper window.
De-convolution of instrument response
In several of instances the seismic information examined didn't, later action without devices de-convolution create marked dissimilarities in outcomes when processed with devices de-convolution.
This method includes the following,
1. Time domain de-convolution of devices reply, applying Several mapping
Frequency domain de-convolution of instrument response
To obtain estimates of the ground acceleration from the recorded relative displacement response, an instrument correction can be applied as
Â where Î³ is the viscous damping ratio, Ï‰ is the transducer's natural frequency and ag (t) is the ground acceleration. The above expression (1) can be used to deconvolve the recorded motion from the ground acceleration in either the time  or frequency domain [4, 7].
3. De-noising methods
Some of the de-nosing techniques are
1. De-noising using a band-pass filtering method
2.De-noising using wavelets
The stationary wavelet transform (SWT) (translation invariant discrete wavelet transform (DWT))
There is however a problem with the wavelet transforms; the discrete wavelet transform (DWT) is not translation invariant [15,16,17]. The coefficients of the discrete wavelet transform (DWT) don't change with a signal; this denotes that the signal is no more orthogonal to most of the fundament functions. More coefficients would be essential to define the signal and the coefficient dimensions would also be much simpler reducing the efficiency of any de-noising scheme.
The report has represented that the execution of the translation invariable wavelet transform (stationary wavelet transform), in the adjustments of seismic information has yielded certain important outcomes. The de-noising of seismic information applying the stationary wavelet transform takes away only those signals whose amplitudes are under some threshold and isn't therefore bandwidth selective.Stationary wavelet transform de-noising prevents the requirement to correct filter cut-off's to fit specified seismic events and is computationally proficient.
Using the Method of Total Least Squares for Seismic Correction:
The benefit of a (LSB) least squares based technique of de-convolution of an devices outcome [1,2,3,4] from seismic information, is that it doesn't need any data about the devices; it only needs the registered seismic information.
A specific trouble in seismic correction techniques is that rather frequently the transfer application of the recording devices isn't recognized. Where device attributes are allowed, a second order single-degree-of-freedom transfer application is used in either the time period or bandwidth domain [8,9,10,11] in order to de-couple the device outcome. Efficient techniques applied to find out a calculation of the inverse filter coefficients are the RLS algorithmic program and it's more stable variant, the square-root recursive least squares algorithmic program [12, 13].
The outcomes validate the importance of applying this technique of de-convolving the device outcomes. These are revealed for 4 kinds of cat's-paw, the SMA-1, the A- 700, the DCA-333 and the SSA-1, which were applied in different Icelandic seismic Issues [19, 20].
However, computationally the Total least Squares algorithmic program needs a large quantity of computer memory when functioning with long information sets and in double precision. Nevertheless, off-line the Total least Squares allows a reasonable instruments for de-convolving the devices outcome allowing an inverse filter as good like if not better to that of the QR-RLS and the second order single-degree-of-freedom, offering a means of de-coupling the devices to get an calculate of the ground motion.
The outcomes represent that the Total least Squares algorithmic program is a valuable tool for setting seismic information when devices attributes are not known. All that is needed is the real recording from the seismograph and the algorithmic program can then bring out the inverse filter with which to de-convolve the devices reply.
The algorithmic program was examined applying information from four devices kinds and was found to be in good arrangement in entire instances but one (Fig 4) with the QR-RLS and the second order SDOF responses. This suggests that the TLS may not be as robust as the QR-RLS in securing the devices outcomes. Nevertheless, the inverse FIR filter diagrams shown are credible outcome and establish the utility of the method. Indeed the Total least Squares Execution in certain instances has been better than that of the QR-RLS and the second order single-degree-of-freedom with a default filter, because it performs the anti-alias filter whose information were in this instance accessible in the record. The sequence of correction is (a) de-trending (b) wavelet de-noising (c) instrument Deconvolution (using the QR-RLS, the Total least Squares or the second order single-degree-of-freedom) (d) fourth order Butterworth filtering, but only in the case of the second order, single-degree-of-freedom process (e) Study plots of proper time-history. The object being to demonstrate that the Total least Squares are an important tool in de-convolving the devices outcome from just the registered Information, when devices attributes are not obtainable.
Inverse filtering using the RLS algorithmThe generic algorithm for the inverse filter problem is shown below in Figure 1. This is same for all adaptive algorithm.It has minimun of one unknown system cascaded with an particular Adaptive algorithm.the solution converges to the inverse of the unknown system. The delay is added to keep the system casual so that the input data, s(n), has sufficient time to reach the adaptive filter
The RLS algorithm can be considered in terms of a least squares solution  of the system of linear equations Ah = d, where rank A is n, the number of unknowns. The objective is to find the vector (or vectors) h of filter coefficients which will satisfy equation (1). This has the well-known solution equations (2) and (3).
However in order to obviate the inverse autocorrelation matrix P, the RLS algorithm provides an efficient method of updating the least squares estimate of the inverse filter coefficients as new data arrive. This is shown in the expression (4).
Where the matrix A is replaced by a single data row, u, and dk forms the desired signal. The updated value of the filter coefficient hk is obtained by adding to the previous value, the 2nd term on the right, which can be considered as a "correction term". The term in brackets is the a priori estimation error defined by (5)
The 2nd term on the right of equation (5) represents an estimate of the desired signal, based on the previous least squares estimate of the filter coefficient. The inverse autocorrelation matrix Pk can be evaluated using Woodbury's identity, which provides an efficient method of updating the matrix, once initialised with an arbitrary value. The update is given in (6) where Î» is the forgetting factor.
Equations (4), (5) and (6) form the basis of the RLS algorithm used in order to obtain the inverse filter coefficients with which to de-convolve the instrument response. The derivation of this algorithm depends on the use of an orthogonal triangulation process known as QR decomposition.
Where 0 is the null matrix, R is upper triangular and Q is a unitary matrix. The QR decomposition of a matrix requires that certain elements of a vector be reduced to zero. The QR-RLS is as follows in equation (8).
where P = the inverse correlation matrix,
Î» = forgetting factor,
Î³ = a scalar and thegain vector is determined from the 1st column of the post-array.
G(n)= is a unitary rotation which operates on the elements of Î»-1/2uH(n)P1/2(n-1) in the pre-array zeroing out each one to give a zero-block entry in the post-array. The filter coefficients are then updated in equation (9) which is the gain vector. This is followed by equation (10) the a priori estimation error.the equation (9),(10),(11)
This is turn, leads to the updating of the least-squares weight vector, h(n), in equation (11). These inverse-filter weights are then convolved with the original seismic data in order to obtain an estimate of the true ground motion.
A comparison of results for seismic data de-convolved using both the LMS and square root RLS for the El-Centro 1940 event, the Sitka 1972 Alaskan event and the Garvey Reservoir event can be shown below. Both does not show a marked difference in the results, at low frequencies.
Figure (1), Sitka event, Alaska, 1972: Power, Phase and Response Spectra
In the Figure (1) , frequency response showed that for higher frequencies different, Amplitudes are too small at the higher frequencies in order to observe any measurable effect. In this case instrument data was available and given as 0.049 for the period and damping 0.570. This was applied to the correction procedure using the standard 2nd order SDOF correction and compared with the results obtained from the QR-RLS and LMS correction. The % increases in the acceleration response spectrum at some structural frequencies is shown below in table 2
% increase in
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 10.0
Table 2, Percentage increase in total acceleration due to RMS/LMS algorithm
RESULTS AND DISCUSSIONS:
The plots of Figure 2, show frequency and phase profiles of two inverse filterderived from the data from the El-Centro 1938 and 1940 seismic events.
FIGURE(2)-Theoretical and RLS inverse filter frequency response profiles for The EL-Centro events
The figure show the after the data was wavelet de-noised frequency responses. both the El-Centro RLS inverse filters and the theoretical responses show an approximately flat response (0dB) in the region of interest. After the region greater than 10Hz the responses differ slightly but again are in general consistent with theory, . The gains vary between approximately 40-120dB, with the theoretical gain at approximately 80dB at half the sampling rate.
Figure(3) Phase plots for the 1938 and 1940 El-Centro seismic events showing almost ero-phase distortion as compared with the theoretical results
The theoretical phase plot in Figure 3 , demonstrate that its inverse filter impresses a phase response, which varies to approximately 100 degrees at 10Hz. phase plots shows that the 1940 El-Centro event exhibits almost zero-phase in the instrument response range up to approximately 10Hz. . This changes with the application of the RLS algorithm.
Figure(4) Figure 4, El-Centro East component magnitude and phase plots using the RLS algorithm post- de-noising
Figure 4 shows the East component of the seismic event, the magnitude plot has a high-pass characteristic though somewhat different in magnitudes at frequencies up to approximately 10Hz. the 1940-E component results would produce a zero-phase distortion but quite different amplitude output to that obtainable by using theoretical methods.Figure4,figure 5 are two sets of plots for the El-Centro N and E components, where RLS de-convolution is performed before wavelet de-noising.
Figure (5), El-Centro North component, magnitude and phase plots using the RLS algorithm post- de-noising
Further magnitude and phase plots are shown in Figure 6 and Figure 7 for the Garvey Reservoir. Figure 6 shows up to approximately 20Hz the magnitude response is almost flat, though one component is shifted by approximately 5dB up to 20Hz. Similarly the phase response is almost flat, though again one component is shifted by 200 degrees. On the other hand Figure 7 gives in the low frequency range up to 20Hz gives a flat magnitude response at almost 0dB and almost zero phase distortion in the same range.
Figure (6), Inverse filter response plots for the Garvey Reservoir where deconvolution is preceded by wavelet de-noising
The above figure shows good result because the actual ground motion is equal to the instrument response..From this we can conclude that the seismic data was takes place in this result in the range of interest de-convolution first and de-noising performing later. ,
Figure 7, Inverse filter response plots for the Garvey Reservoir where de convolution occurs before wavelet de noising.
Novel Seismic Correction approaches without instrument data, using adaptive methods and De-Noising.
13th World Conference on Earthquake Engineering Vancouver, B. C., Canada August 1-6, 2004 Paper No. 2664
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