**Authors**

**Abstract**

Temporal resolution of seismic data is proportional to the seismic band width. Seismic data still have not enough temporal resolution because of the band-limited nature of available data even if it is deconvolved. Lower and higher frequencies of seismic data spectrum are missing and cannot be recovered by the usual deconvolution methods. Because of absorption, high frequencies belonging to the spectrum are missing and recovery of lower frequencies is also a big deal (Lindseth, 1979). Many different deconvolution techniques have been developed to process the data obtained from various sources ranging of seismic data. especially, since for many years in seismic processing, they have been used to improve the temporal resolution of seismic data. In this paper we introduce a method that is the generalization of the autoregressive (AR) spectral extrapolation based method originally applied by Hakan Karsli (2006), which extrapolates the deconvolved seismic spectrum for recovery of missed frequencies. When reflectors are numerous, the seismic spectrum is complicated and extrapolation by AR-based methods is uncertain. The introduced method takes a certain part of both real and imaginary parts of the spectrum, where S/N is high compare to the rest of the spectrum, and extrapolates lower and higher portions of the spectrum using Singular Spectrum Analysis (SSA) and Autoregressive model. Experience shows that a 3–10 dB drop from the maximum amplitude of the spectrum of the source wavelet represents a high SNR portion of the spectrum. Because of the existence of unwanted noise, the usual regression algorithms do not lead to favorable results. In second step of extrapolation algorithm we decompose selected spectrum by SSA.

SSA is a tool to extract information from short and noisy chaotic time series (Vautard et al., 1992). It relies on the Karhunen-Loeve decomposition of an estimate of the covariance matrix based on "M" lagged copies of the time series. Thus as the first step, the embedding procedure is applied to construct a sequence of M-dimensional vectors from the time series:

The N' × M trajectory matrix (D) of the time series has the M dimensional vectors as its columns, and is obviously a Hankel matrix (the elements on the diagonals j + j = constant are equal). In the second step, the M × M covariance matrix is calculated as:

Eigen elements can be determined by Singular Value Decomposition (SVD):

The elements of diagonal matrix ?= [diag (?1. . . ?M)] are the singular values of D and are equal to the square roots of the Eigenvalues. The Eigen elements {( , ): k = 1. . . M} are obtained from:

Each Eigenvalue, estimates the partial variance in the direction, and the sum of all Eigenvalues equals the total variance of the original time series. In the third step, the time series is projected onto each Eigenvector, and yields the corresponding principal component (PC) for each (t):

Each of the principal components, a nonlinear or linear trend, a periodic or quasi-periodic pattern, or a multiperiodic pattern, has a narrow band frequency spectrum and well defined characteristics to be estimated.

As the fourth step, the time series is reconstructed by combining the associated principal components:

Data extrapolation algorithms based on AR techniques have been commonly used for modeling the past values (backward) and future values (forward) of a signal Walker and Ulrych, 1983; Miyashita et al., 1985; Each principal component is applied to the AR extrapolation method, to obtain the next and previous missed frequencies of that principal component using the following extrapolation equation:

That is the autoregressive equation of order p, i.e. extrapolate kth frequency based on the linear sum of p previous frequencies. are AR coefficients and is random noise. The coefficients can be computed from autocorrelation estimates, from partial autocorrelation, and from least-squares matrix procedures. There are several approaches to select the model order for practical situations.

In this study, AR model order L is selected equal to 0.3 times of the length of the high S/N portion of the trace spectrum, which is suggested by Walker and Ulrych (1983).

After extrapolation of each principal component, the trace spectrum is reconstructed by combining the associated extrapolated principal components. The seismic data whose temporal resolution has been improved is calculated by an inverse Fourier transform of SSA and AR spectral extrapolated spectrum. The results from synthetic and real seismic data are presented.

SSA is a tool to extract information from short and noisy chaotic time series (Vautard et al., 1992). It relies on the Karhunen-Loeve decomposition of an estimate of the covariance matrix based on "M" lagged copies of the time series. Thus as the first step, the embedding procedure is applied to construct a sequence of M-dimensional vectors from the time series:

The N' × M trajectory matrix (D) of the time series has the M dimensional vectors as its columns, and is obviously a Hankel matrix (the elements on the diagonals j + j = constant are equal). In the second step, the M × M covariance matrix is calculated as:

Eigen elements can be determined by Singular Value Decomposition (SVD):

The elements of diagonal matrix ?= [diag (?1. . . ?M)] are the singular values of D and are equal to the square roots of the Eigenvalues. The Eigen elements {( , ): k = 1. . . M} are obtained from:

Each Eigenvalue, estimates the partial variance in the direction, and the sum of all Eigenvalues equals the total variance of the original time series. In the third step, the time series is projected onto each Eigenvector, and yields the corresponding principal component (PC) for each (t):

Each of the principal components, a nonlinear or linear trend, a periodic or quasi-periodic pattern, or a multiperiodic pattern, has a narrow band frequency spectrum and well defined characteristics to be estimated.

As the fourth step, the time series is reconstructed by combining the associated principal components:

Data extrapolation algorithms based on AR techniques have been commonly used for modeling the past values (backward) and future values (forward) of a signal Walker and Ulrych, 1983; Miyashita et al., 1985; Each principal component is applied to the AR extrapolation method, to obtain the next and previous missed frequencies of that principal component using the following extrapolation equation:

That is the autoregressive equation of order p, i.e. extrapolate kth frequency based on the linear sum of p previous frequencies. are AR coefficients and is random noise. The coefficients can be computed from autocorrelation estimates, from partial autocorrelation, and from least-squares matrix procedures. There are several approaches to select the model order for practical situations.

In this study, AR model order L is selected equal to 0.3 times of the length of the high S/N portion of the trace spectrum, which is suggested by Walker and Ulrych (1983).

After extrapolation of each principal component, the trace spectrum is reconstructed by combining the associated extrapolated principal components. The seismic data whose temporal resolution has been improved is calculated by an inverse Fourier transform of SSA and AR spectral extrapolated spectrum. The results from synthetic and real seismic data are presented.

**Keywords**