# Data Analysis Using Total Least Square Algorithm Computer Science

Seismology is the study of under Earth by measuring vibrations on the Earth's surface. Seismology can either be passive or active. The passive is by just listening to vibrations caused by earthquakes and volcanic activity where as the active one is by using small explosive charges to send vibrations into the ground. Today, most of the seismograms are recorded digitally to make analysis easily by the computers. Seismograms are essential for measuring earthquakes using the Richter scale. High fidelity digital accelerometers, analogue instruments are utilizing to obtain the ground motion data during earthquakes. Analogue one could be compared with digital one. Analogue one is considerably cheaper to manufacture and maintain. The dynamic behavior of structures during earthquakes understands by data from the former needs to be processed. The digital one is being expensive to maintain.The benefit of digital instruments is that the determinations of accurate results as compared to analogue one for utilize in earthquake studies.

An accelerometer is used on a seismograph to record acceleration as utility of time. The recorded data is really the response of the instrument to ground motion before precise motion of the ground itself. Moreover, the data recorded is certainly noisy due to several factors like ocean waves, heavy traffic, piling and wind in equally low and high frequency range. In addition to the above is the truth that the largest part of the historical data that exist today is from records of analogue instruments of unidentified characteristics or questionable reliability. It is therefore very important to route the recorded information by means of digital filtering techniques to recover the data that define ground motion to the extent feasible.

Earthquake engineering has developed a lot recently and most of the complex designs now use special earthquake protective elements either just in the foundation (base isolation) or distributed throughout the structure. The analysis depends on the structure of the building or an object which is considered for the analysis. There are various parameters that are to be kept in notice while analyzing the seismic data for these structures. Analyzing these types of structures requires specialized explicit finite element computer code.

Structural analysis methods can be divided into the following five categories.

Equivalent Static Analysis

Response Spectrum Analysis

Linear Dynamic Analysis

Non-linear Static Analysis

Non-linear Dynamic Analysis

**Equivalent Static Analysis**

This method of analysis explains a series of forces acting on a building to represent the effect of earthquake ground motion, typically defined by a seismic design response spectrum. It assumes that the building responds in its fundamental mode. For this to be true, the building must be low-rise and must not twist significantly when the ground moves. The response is read from a design response spectrum, given the natural frequency of the building (either calculated or defined by the building code). To account for effects due to "yielding" of the structure, many codes apply modification factors that reduce the design forces.

**Response Spectrum Analysis**

This approach mainly depends up on the multiple modes of response of a building to be taken into account (in the frequency domain). This method is applicable for many building codes for all except very simple or complex structures. Computer analysis can be used to determine these modes for a structure. The response of a structure can be defined as a combination of multiple modes in a vibrating string which correspond to the "harmonics". While analyzing the seismic data for each and every mode, a response is obtained from the design spectrum (which is based on the modal frequency and the modal mass).These two values are then combined to provide an estimate of the total response of the structure.The result of a response spectrum analysis using the response spectrum from a ground motion is typically different from that which would be calculated directly from a linear dynamic analysis using that ground motion directly, since phase information is lost in the process of generating the response spectrum.For irregular structures like very simple and complex, this method is not applicable and therefore complex analysis is often required, such as non-linear static or dynamic analysis.

**Linear Dynamic Analysis**

The Static procedures are not applicable for higher mode effects. Therefore we have to go for high end analysis procedures like Linear Dynamic Analysis. The advantage of these linear dynamic procedures with respect to linear static procedures is that higher modes can be considered. The seismic input is modeled using either modal spectral analysis or time history analysis. In both the cases, the corresponding internal forces and displacements are determined using linear elastic analysis. In this Analysis the applicability decreases with increasing nonlinear behavior. In linear dynamic analysis, the response of the structure to ground motion is calculated in the time domain, and all phase information is therefore maintained. Only linear properties are assumed.

**Non-linear Static Analysis**

The Non-linear static analysis is used to reduce the uncertainty and conservatism. In general, the linear procedures are applicable for the structures which remains almost elastic for the level of ground motion or uniform distribution of nonlinear response throughout the structure. As the performance implies great inelastic demands, the uncertainty of the linear procedures also increased. Here the procedure requires high level of conservatism to avoid unintended performance. Therefore, procedures incorporating inelastic analysis can reduce the uncertainty and conservatism.

The Non-linear static approach is also known as "pushover" analysis. Here a pattern of forces is applied to a structural model that includes non-linear properties (such as steel yield), and the total force is plotted against a reference displacement to define a capacity curve. This can then be combined with a demand curve (typically in the form of an acceleration-displacement response spectrum (ADRS)). This essentially reduces the problem to a single degree of freedom system.

**Non-linear Dynamic Analysis**

Nonlinear dynamic analysis utilizes the combination of ground motion records with a detailed structural model, therefore is capable of producing results with relatively low uncertainty. In nonlinear dynamic analysis, the detailed structural model subjected to a ground-motion record produces estimates of component deformations for each degree of freedom in the model and the modal responses are combined using schemes such as the square-root-sum-of-squares. This approach is the most rigorous, and is required by some building codes for buildings of unusual configuration or of special importance.

**ABSTRACT**

The main objective of this project is the implementation of TLS (Total Least Square) algorithm and to discuss a relatively straight forward approach in the context of a system identification problem. This project describes the correction or recovery of the original ground motion acceleration time histories from accelerometer digital records. It deals specifically with the situation where the recording accelerometer instrument is unknown. Several adjustments are to be done for raw accelerogram data. To perform this operation a device called Adaptive filter can be used .In performing the correction of accelerogram data MATLAB code plays an important role .The project also discusses the order in which the implementation of the TLS algorithm should be applied. Total least squares is also known as errors in variables, rigorous least squares, or orthogonal regression. This least squares data modeling technique take observational errors on both dependent and independent variables into account. It can be applied to both linear and non-linear models.

The total least squares (TLS) method is used to identify the unknown system (instrument) that must be used to de-convolute the recorded time histories. After comparing and contrasting this method with the recursive least squares method (RLS) and a standard second order, single-degree-of-freedom, idealized instrument de-convolution it is proved that TLS provide a reasonable estimate of its characteristics from just the recorded historical data without any assumed information about the instrument.

**LITERATURE REVIEW**

**Adaptive filter:**

An adaptive filter is a filter that self-adjusts its relocate utility according to an optimizing algorithm. Because utilization of the complexity of the optimizing algorithms, most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signal. By way of disparity, a non adaptive filter has static filter coefficients (which together form the relocate utility).For a few applications, adaptive coefficients are requisite since a few parameters of the much loved processing business (for illustration, the properties of a few sound signals) are not known in move on. In these situations it is common to employ an adaptive filter, which uses feedback to refine the values of the filter coefficients and hence its frequency retort. Generally speaking, the adjusting process engross the utilization of a cost utility, which is a measure for optimum performance of the filter (for example, minimizing the sound constituent of the input), to feed an algorithm, which determines how to modify the filter coefficients to minimize the cost on the next iteration.

The two most widely used adaptive techniques are Least Mean-square (LMS) and the Recursive Least-Square(RLS). Least Mean Square (LMS) error and Recursive Least-Square (RLS) error techniques are mostly used adaptive techniques for instrument correction of the accelerogram data. LMS algorithms adapt the filter coefficients at every iterative step to minimize a cost function that differs from one variant of LMS to another. LMS algorithms are simpler to implement, do not involve any matrix operations and are hence faster compared to RLS techniques. Minimizing with respect to the mean-square error produces the same set of filter coefficients for all sequences which have the same statistics, the coefficients depend on the ensemble average of the data rather than on the data itself. With the least squares error, the squared error is minimized with an explicit dependence on the values of the data itself i.e. x(n). This in turn means that for different sets of signal data we obtain different filter coefficients, even if the statistics of the data sequences considered are the same. Therefore a RLS minimized data set yields a set of filter coefficients which will be optimal for a given set of data, instead of being statistically optimal over some particular event. It is considered that an RLS adaptive predictor is best suited to non-stationary seismic events.[16,17]

Data correction techniques based on minimizing some cost function of Least Squares Error do not require instrument characteristics information [1, 3, 7, 8] and relies only on the recorded accelerogram data. This is the great advantage with which deconvolution of seismic data is possible. Many techniques for data correction assume a second order SDOF instrument model which is deconvolved with instrument response to obtain an estimate of actual ground motion. [3] discusses the development of recursive least squares (RLS) algorithm for system identification. Denoising is done by means of stationary wavelet transformation (SWT) rather than high pass filtering to reduce artifacts.

Chanerley et al. in [9] estimates the bispectra of seismic data using a nonlinear model. We have to estimate the parameters of an approximate linear model first using linear predictive coding and then the transformation of linear model to time domain to remove linear component from the data is done. The authors in [10] describe implementation of Total Least Squares (TLS) algortithm minimizing || Ax - b|| where A is the corrected data matrix and b is the error vector. This difference in TLS is called partial TLS and is investigated in [12]. The TLS algorithm assumes that the part of data obtained from the instrument is known exactly while the rest is noisy. PTLS was used for denoising and correcting the baseline error without any frequency selective filters. SWT was used instead of band-pass filters in order to get a better ground motion data estimate[7]. The recursive least squares (RLS) algorithm was used for evolving an inverse filter, which is applied to deconvolve the seismic data[3]. RLS was considered to be the preferred choice over LMS when instrument data isn't available in [8]and as the resulting adaptive filter is optimal for the given set of data rather than for an ensemble average. It also converges faster compared to LMS algorithm [3]. Chanerley and Alexander in [11] apply Lp optimisation with iterative least squares technique to determine the sensitivity of the algorithm to bursts of short-duration large-amplitude noise which typically unstable instrument operation.

The correction techniques are used to digitize the analog seismic data recorded by analogue instruments. They also correct the data for instrument characteristics, detrend and denoise, and resample to an appropriate sampling rate.

Attempts have been made to devise correction procedure for accelerogram data in 1970 by Trifurnac et al. [13]. Here a low-pass filtered is used initially then the corrected instrument data will be passed on to the high-pass filtering to get rid of baseline error. He used an FIR filter (Ormbsy), which is known for phase distortion. Converse [14] had introduced a computer program (BAP) in 1991, which involved in resampling by interpolation to 600Hz. Here initially the baseline error was removed and then the instrument correction will be done. Thereafter the data will be passed through a high-pass bi-directional IIR butterworth filter for denoising. BAP will run the algorithm segment wise on the data rather than whole record at a time which will make the computing easy[1].

There are several steps involved in correction procedures. Some of them are resampling, baseline correction, instrument correction, filtering & phase correction, decimation are described in [1]. Resampling is done in order to make the accelerogram data sampled evenly. As per the latest technology even high sampling rates can be conveniently handled without segmentation. Even then the dynamic response of instruments is reflected in seismic data. Hence, a correction is necessary in order to have a better estimate of actual ground motion. Filtering plays a major role eliminating the external source noise. Decimation technique is used to down sample the frequency in order to reduce the processing time. Decimation involves in down sampling and low pass filtering, which prevents aliasing.

The slightest signify squares (LMS) algorithms regulate the filter coefficients to minimize the cost utility. Compared to recursive slightest squares (RLS) algorithms, the LMS algorithms do not involve any matrix operations. So, the LMS algorithms need fewer computational assets and memory than the RLS algorithms. The implementation of the LMS algorithms also is less complicated than the RLS algorithms. TLS may not be as robust as the QR-RLS in securing the instrument response. However, the eigenvalue spread of the input correlation matrix, or the correlation matrix of the input signal, might affect the convergence speed of the resulting adaptive filter.

This paper explores instrument correction of accelerogram data using a Total Least Square Algorithm.

**ALGORITHM**

**Total Least Squares algorithm**

Total least squares algorithm, which is also known as errors in variables, rigorous least squares, or orthogonal regression, is a least squares data modeling algorithm in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression, and can be related to both linear and non-linear models[20].

Total Least Squares (TLS) is an extension of the usual Least Squares method: it permits dealing also with uncertainties on the sensitivity matrix.

**Brief Introduction on Least Squares**

Least squares (LS) problems are those in which the intented function may be stated as a sum of squares. Such problems have a natural relationship to distances in Euclidean geometry, and the solutions may be calculated methodically using the tools of linear algebra.

**Regression**

Least Squares regression is the most fundamental form of LS optimization problem. Suppose given a set of measurements, yn accumulted for different parameter values, xn. The purpose of LS regression problem is to find:

The above expression can be re written in terms of column N-vectors as:

Now to obtain the solution we can describe three ways. The conventional (non-linear-algebra) approach is to use calculus. With respect to p if we set the derivative of the expression equal to zero and solve for p, we get:

In technical point of view, one should check that this is a minimum (and not a maximum or saddle point) of the expression. But since the expression is a sum of squares, we identify the solution must be a minimum

A second method of getting the solution comes from taking into account the geometry of the problem in the N-dimensional space of the data vector. We ask for a scale factor, p, such that the scaled vector is as close as possible (in a Euclidean-distance sense) to .Geometrically, we know that the scaled vector must be the projection of against the line in the direction of

Thus, the solution for p is the same as below.

A third and last method of getting the solution comes from the so-called orthogonality principle. The idea is that the error vector for the optimal p should be perpendicular to

Solving for p gives the same result as below.

**Total Least Squares (Orthogonal) Regression**

In traditional least-squares regression, errors are described as the squared distance from the data points to the fitted function, as calculated along a particular axis direction. But if there is not a understandable assignment of "dependent" and "independent" variables, then it makes more sense to calculate errors as the squared perpendicular distance to the fitted function. The main disadvantage of this formulation is that the fitted surfaces must be subspaces (lines, planes, hyper planes).

Suppose one needs to fit the N-dimensional data with a subspace (line/plane/hyper plane) of dimensionality N âˆ’ 1. The space is suitably defined as consisting of all vectors perpendicular to a unit vector and the optimization problem may thus be expressed as:

where M is a matrix consisting the data vectors in its rows.

Performing a Singular Value Decomposition (SVD) on the matrix M permits us to find the solution very easily. In exact, let with U and V orthogonal, and S diagonal with positive decreasing elements. Then

As V is an orthogonal matrix, we can alter the minimization problem by substituting the vector which has the identical length as :

The matrix is square and diagonal, which has diagonal entries As a result of this the expression, which is being minimized is a weighted sum of the components of and must be greater than the square of the smallest (last) singular value, :

where in the last step we have used the constraint that is a unit vector. In addition, the above equation becomes an equality when which is the standard basis vector associated with the Nth axis.

To get a solution for we can transform this solution back to the original coordinate system

which is represented as the Nth column of the matrix V . To be brief, the minimum value of the equation occurs when we set equal to the column of V associated with the minimal singular value.

The formulation can easily be improved to include a shift of origin. This means, suppose we wish to fit the data with a line/plane/hyper plane ,which does not necessarily pass through the origin:

where represents a column vector of ones. For a known the optimal solution for u0 is easily established to be where is a vector whose components are the average of each column of M

Suppose we wanted to fit the data with a line/plane/hyper plane of dimension Nâˆ’2? First we shouldt find the direction along which the data vary least, project the data into the remaining (N âˆ’ 1)-dimensional space, and then repeat the process. As V is an orthogonal matrix, the secondary solution will be the second column of V (i.e., the column associated with the second-largest singular value). In common, the columns of V offer a basis for the data space, in which the axes are ordered according to variability. We can work out for a vector subspace of any desired dimensionality in which the data are closest to lying.

The total least squares problem may also be derived as a pure (unconstrained) optimization problem using a form known as the Rayleigh Quotient:

The length of the vector doesn't change the value of the fraction, so one usually solves for a unit vector. From the above expressions, this fraction takes on values in the range which is equal to the minimum value when the first column of the matrix V.

In the domain of uncertainty of the sensitivity matrix the TLS problem is an addition of the LS problem. The TLS solutions are less stable than the usual LS ones, the deep reason being that while the LS problem has always a solution, the TLS problem doesn't have, but the reason why the TLS is a suitable method of analysis of linear problems (and nonlinear, with an iterative approach) is that it gives the opportunity to explore the validity of the model used, giving a measure of the change in the sensitivity matrix needed to have the best solution. Hence the TLS can give the scientist hints for recognizing mistakes in the model, while in the case of LS recognizing mistakes is difficult to work. Therefore the TLS is more suitable since conceptually more correct in the cases where the model used is really empirical and it's a priori scientific motivations are not strong [18].

**ANALYSIS OF GRAPHS**

**005257XA.RAW**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005257XA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable,from 42-68Hz the it takes cosine form then finally raises exponentially.

LINEAR

The graphs represent different types of responses of seismic data using 005257XA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spike occurs nearly at 5Hz after that it fluctuates. The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-40Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005257YA.RAW raw data, except that the plot was taken in logarithmic form.

**005257YA.RAW:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005257YA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable,from 42-61Hz the it takes cosine form then finally raises exponentially.

LINEAR:

The graphs represent different types of responses of seismic data using 005257YA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 3Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005257YA.RAW raw data, except that the plot was taken in logarithmic form.

**005257ZA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005257ZA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable,from 42-68Hz the it takes cosine form then finally raises exponentially.

LINEAR:

The graphs represent different types of responses of seismic data using 005257ZA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 7Hz after that it fluctuates.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005257ZA.RAW raw data, except that the plot was taken in logarithmic form.

**005267XA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005267XA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-20Hz the frequency response is stable,from 20-73Hz the it increases sinusoidally then finally raises exponentially.

LINEAR:

The graphs represent different types of responses of seismic data using 005267XA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 10Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005267XA.RAW raw data, except that the plot was taken in logarithmic form.

**005267YA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005267YA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-20Hz the frequency response is stable,from 20-73Hz the it increases sinusoidally then finally raises exponentially.

LINEAR:

The graphs represent different types of responses of seismic data using 005267YA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 9Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-25Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005267YA.RAW raw data, except that the plot was taken in logarithmic form.

**005267ZA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005267ZA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-20Hz the frequency response is stable,from 20-73Hz the it increases sinusoidally then finally raises exponentially.

LINEAR:

The graphs represent different types of responses of seismic data using 005267ZA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 13Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005267ZA.RAW raw data, except that the plot was taken in logarithmic form.

Linear:000157XA

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 000157XA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-25Hz the frequency response is stable,from 25-70Hz the it increases sinusoidally then finally raises exponentially.

The graphs represent different types of responses of seismic data using 000157XA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that there is a fluctuation in the acceleration in the frequency range 0-25Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 000157XA.RAW raw data, except that the plot was taken in logarithmic form.

LINEAR(000157YA):

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 000157YA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-25Hz the frequency response is stable,from 25-70Hz the it increases sinusoidally then finally raises exponentially.

The graphs represent different types of responses of seismic data using 000157Y.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that the maximum spikes occurs nearly at 5Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-40Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 000157YA.RAW raw data, except that the plot was taken in logarithmic form.

LINEAR(000157ZA):

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 000157ZA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable,from 40-70Hz the it increases sinusoidally then finally raises exponentially.

The graphs represent different types of responses of seismic data using 000157ZA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that there is a fluctuation in the acceleration in the frequency range 0-25Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-40Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 000157ZA.RAW raw data, except that the plot was taken in logarithmic form.

**005238XA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005238XA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable, then finally raises exponentially

LINEAR:

The graphs represent different types of responses of seismic data using 005238XA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-20Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that there is a fluctuation in the acceleration in the frequency range 0-25Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005238XA.RAW raw data, except that the plot was taken in logarithmic form.

**005238YA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005238YA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-50Hz the frequency response is stable, then finally raises exponentially

LINEAR:

The graphs represent different types of responses of seismic data using 005238YA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-40Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that there is a fluctuation in the acceleration in the frequency range 0-25Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-30Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005238YA.RAW raw data, except that the plot was taken in logarithmic form.

**005238ZA:**

The above graphs represents the inverse filter frequency response of the instrument from which the seismic data is collected.The frequency response was taken in linear form using the raw data 005238ZA.raw.It is a plot of frequency vector versus frequency response where X-axis represents the frequency vector and Y-axis represents frequency response in db.It can be observed from graph that from 0-40Hz the frequency response is stable, then finally raises exponentially

LINEAR:

The graphs represent different types of responses of seismic data using 005238ZA.RAW raw data. First graph represents seismographs acceleration as a function of time. Second graph represents Welch's power spectral density where X-axis represents frequency and Y-axis represents PSD in db. It can be observed that in the frequency range 0-50Hz there is raise in the PSD .Third graph represents acceleration response spectrum calculated from time history using transfer function. It can be observed from the graph that there is a fluctuation in the acceleration in the frequency range 0-25Hz.The last graph represents phase spectrum in terms of frequency and unwrapped phase angle. It can be observed that in between the frequency range 0-75Hz the phase angle decreases, after that it becomes stable.

LOG:

The above graphs are same as previous graphs which represent different types of responses of seismic data using 005238ZA.RAW raw data, except that the plot was taken in logarithmic form.

**CONCLUSION**

This dissertation "Analysis of seismic data using TLS Algorithm" yields the results that the TLS algorithm is a useful tool for correcting seismic data when instrument parameters are not known. All that is required is the original recording from the seismograph and the algorithm can then produce the inverse filter with which we can de-convolve the instrument response. The algorithm was tested using data from various instruments and found that the TLS may not be as robust as the QR-RLS in securing the instrument response. The inverse FIR filter plots shown are credible responses and explains the utility of the approach. In fact the TLS performance in some cases has been better than that of the QR-RLS and the 2nd order SDOF with a standard filter, because it reflects the anti-alias filter whose details were in this case available in the record. In general the TLS algorithm demonstrates that it can be used effectively to deconvolve the instrument response from the seismic data, in particular where the instrument parameters are either not known or not available. Considerably the TLS algorithm requires a large amount of memory when working with large data sets and in double precision.

The TLS approach is an advanced to LS approach. The TLS solutions are less stable than the usual LS solutions. The reason why the TLS is a suitable method for analysis of linear and nonlinear problems is that, it gives the flexibility to find the validity of the model used. This is done by providing a measure of manipulating the values of the sensitivity matrix to obtain the best solution. Therefore the TLS helps scientists in identifying the mistakes in the model where this is not the case in LS.

Article name: **Data Analysis Using Total Least Square Algorithm Computer Science essay, research paper, dissertation**