The general principle of fourier regularization is to compute the spectrum of irregularly sampled data, and then use an inverse transform to reconstruct data at new locations. Unlike resolution, leakage can be reduced by increased time domain sampling because of the increased separation between spectral replicas. Both resolution and leakage depend on the shape of the data window. Breve introduzione al mondo dellelaborazione dei segnali in ambito digitale. Sampling, for instance, produces leakage, which we call aliases of the original spectral component. Michel schonewille, andreas klaedtke, alan vigner, john brittan and tony martin present a solution to a problem with regularly sampled data found in the use of the anti alias anti leakage fourier transform alft. Pdf antileakage fourier transform for seismic data regularization. Deriving fourier transform from fourier series consider a periodic signal f with period t. A first, standard anti leakage fourier transform method is applied to unaliased frequency components of temporaltransformed seismic data and, a second, nonstandard anti leakage fourier transform method is applied to aliased frequency components of the temporaltransformed seismic data seismic data. The antileakage fourier transform algo rithm can overcome the spectral leakage problem and han dles aliased data.
The antileakage fourier transform reconstruction xu et al. Antialiasing, antileakage fourier transform earthdoc. Leakage effect occuring when a discrete fourier transform is performed over a hamming time window. Journal of geophysics and engineering, volume, number 1. Thus the ripples of the sinc function will show up as leakage far from any spectral peak that is not perfectly periodic in the ffts length. The application of fourier analysis to forecasting the inbound call time series of a call centre bruce g. Dec 22, 2010 the antileakage fourier transform algorithm can overcome the spectral leakage problem and handles aliased data. An adaptation of this can give a robust solution to fourier regularisation, as discussed by xu and pham 2004a, xu and pham 2004b and schonewille 2009, and has been named the antileakage fourier transform due to its ability to suppress spectral leakage. An estimated frequencywavenumber spectrum is generated by applying a first antileakage fourier transform method to aliased frequency components in temporaltransformed seismic data and applying a second antileakage fourier transform method to unaliased frequency components in the temporaltransformed seismic data.
This wellknown phenomenon is called spectral leakage. The fast fourier transform fft and the power spectrum are powerful tools for analyzing and measuring signals from plugin data acquisition daq devices. Seismic data regularization with antileakage fourier transform. The resulting transform pairs are shown below to a common horizontal scale. Figure 1 schematically describes the main areas of seis mic processing in which interpolation and regularization. The antileakage fourier transform algorithm can overcome the spectral leakage problem and handles aliased data. For fourier transform purposes, sampling is modeled as a product between st and a dirac comb function. For analysis of signals with dft, the length of discrete signal must be finite so signal to be analyzed must be divided to some windows.
Radio spectrum estimates using windowed data and the discrete fourier transform roger dalke. This is an iterative algorithm, which exploits the fact that leakage. To attenuate all aliases and the leakage of this component onto other fourier coefficients, the data component corresponding. However, the computational cost dramatically increases with the sampling dimensions. Digital signal processing algorithms are commonly used to obtain radio spectrum estimates based on measurements. The reverse transform can then be used either to output the data on a regular grid for full regularisation or to. Any other type of operation creates new frequency components that may be referred to as spectral leakage in the broadest sense. Michel schonewille, andreas klaedtke, alan vigner, john brittan and tony martin present a solution to a problem with regularly sampled data found in the use of the. A wider window causes less distortion, but uses older signal data. This method uses discrete fourier transform dft, although for regular data with missing traces or gaps it can use the fast fourier transform fft zwartjes and sacchi 2007. Equation 3 can also be used to obtain the frequency quantum.
However, this is a very sharpened window that causes broad leakage. One of the methods for 5d fourier based seismic interpolation is the antileakage fourier transfrom alft. The anti leakage fourier transform reconstruction xu et al. As a consequence, energy from one fourier coefficient leaks onto other coefficients. In this paper, we present an algorithm, called antileakage fourier transform, for seismic data reconstruction from an irregularly sampled grid to a regular grid that overcomes these difficulties. For example, you can effectively acquire timedomain signals, measure the frequency content, and convert the results to realworld units and displays as shown on traditional benchtop. Matching pursuit fourier interpolation using priors derived. There is also an inverse fourier transform that mathematically synthesizes the original function from its frequency domain representation. Antileakage fourier transform for seismic data regularization. The fourier transform is sometimes denoted by the operator fand its inverse by f1, so that. Its discrete fourier transform is likewise a finite length sequence. Returning to case of the spectrum analyzer, the fourier transform, leakage causes a conflict of interests. Chapter 3 fourier transforms of distributions questions 1 how do we transform a function f. Discrete fourier transform dft is the basic part of various algorithms of signal processing in many fields of science and technology.
Despite the simplicity of the antileakage fourier transform alft, and its ability to regularize pure nonuniform data sets and. We investigate the nonorthogonality of the fourier basis on an irregularly sampled grid and propose a technique called antileakage fourier transform to overcome the spectral leakage. Consequently, spectrum leakage appears in frequency domain. In reality, signals are of timelimited nature and nothing can be known about the signal beyond the measured interval. Radio spectrum estimates using windowed data and the discrete. Mar 09, 2015 leakage effect occuring when a discrete fourier transform is performed over a hamming time window. However, the computational cost dra matically increases with the sampling dimensions. Ep2103959a2 method for interpolating seismic data by anti. Jan 29, 2016 one of the popular seismic interpolation techniques is anti leakage fourier transform alft which was introduced by xu and pham.
Seismic data regularization with the antialias antileakage. Antileakage fourier transform for seismic data regularization free. Im also overjoyed that my method gives identical results to its much speedier cousin, the fast fourier transform fft, because the slow fourier transform sft has a number of very significant advantages over the fft. Despite the simplicity of the antileakage fourier transform alft, and its ability to regularize pure nonuniform data sets and handle higher dimension seismic data, alft still remains less appealing because of its iterative algorithm and so its costly computation. Antileakage fourier transform for seismic data regularization in. Despite the simplicity of the anti leakage fourier transform alft, and its ability to regularize pure nonuniform data sets and handle higher dimension seismic data, alft still remains less appealing because of its iterative algorithm and so its costly computation. Summary the antileakage fourier transform alft is a regularization method using an iterative procedure for computing the spectrum of irregularly. The application of fourier analysis to forecasting the. Introduction to fourier transforms fourier transform as a limit of the fourier series inverse fourier transform.
The key to resolving the spectral leakage is to reduce the leakages among fourier coefficients in the original data before the calculation. Antileakage fourier transform the first documented solution to the leakage problem was performed by hogbom 1974 as a solution within astrophysics. Anti leakage fourier transform the first documented solution to the leakage problem was performed by hogbom 1974 as a solution within astrophysics. The anti leakage fourier transform alft is a regularization method using an iterative procedure for computing the spectrum of irregularly sampled data. Antileakage fourier transform we have developed an algorithm that performs a forward irregular fourier transform in all five data dimensions simultaneously to build a representation of the data in the frequency domain. Antileakage fourier transform alft and its application for seismic data reconstruction article pdf available in journal of the earth and space physics 394. One of the popular seismic interpolation techniques is antileakage fourier transform alft which was introduced by xu and pham. Since each of the rectangular pulses on the right has a fourier transform given by 2 sin ww, the convolution property tells us that the triangular function will have a fourier transform given by the square of 2 sin ww. To generalize it to higher dimensions, we proposeanareaweightingschemetoaccuratelyestimatethe fourier components. Transform each trace from time domain to frequency domain using the fast fourier transform fft generate a data series using the available traces for each frequency a frequency slice use the antileakage lssa to regularize each frequency slice transform each trace from frequency domain to time domain using the inverse fft. Such algorithms allow the user to apply a variety of timedomain windows and the discrete fourier transform to rf signals and noise.
Note that the transform of a rectangular window is a sinc function sinxx, which has infinite width. These advantages are particularly important in climate science. This interspectral mixing is called leakage and is distinct from the loss of resolution that occurs within the spectral replica. Even if the window contains integer multiples of a period erroneous spectral components occur. In this talk first several aspects of the alft are illustrated using simple synthetic signals. Bellc answ police assistance line, tuggerah, nsw 2259, email. The frequency quantum is the change in frequency that is represented by two sequential bucket numbers. Anti leakage fourier transform we have developed an algorithm that performs a forward irregular fourier transform in all five data dimensions simultaneously to build a representation of the data in the frequency domain. Consider this fourier transform pair for a small t and large t, say t 1 and t 5. Then, the maximum fourier component is selected and transformed back to the irregular grid. Pham, 2004, seismic data regularization with antileakage.
The fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. Transform each trace from time domain to frequency domain using the fast fourier transform fft generate a data series using the available traces for each frequency a frequency slice use the anti leakage lssa to regularize each frequency slice transform each trace from frequency domain to time domain using the inverse fft. The inverse fourier transform for linearsystems we saw that it is convenient to represent a signal fx as a sum of scaled and shifted sinusoids. Seismic data regularization with the antialias antileakage fourier. Resolution and leakage of fourier transform concepts and.
Leakage effect of a discrete fourier transform hamming window. Many advantages of regularization in the fourier domain come with some serious problems. Town park drive houston tx 77072 usa in the theory of fourier reconstruction from discrete seismic data it aims to estimate the spatial frequency content on an irregularly sampled grid. Fourier series corresponding to an even function, only cosine terms and possibly a constant which we shall consider a cosine term can be present. To generalize it to higher dimensions, we propose an area weighting scheme to accurately estimate the fourier components. On sampling functions and fourier reconstruction methods. Imaging subsalt structures under complex overburdens is a chal.
An estimated frequencywavenumber spectrum is generated by applying a first anti leakage fourier transform method to aliased frequency components in temporaltransformed seismic data and applying a second anti leakage fourier transform method to unaliased frequency components in the temporaltransformed seismic data. Spectral leakage by douglas lyon abstract this paper is part 4 in a series of papers about the discrete fourier transform dft and the inverse discrete fourier transform idft. D032 seismic data regularization with antileakage fourier transform sheng xu and don pham abstract 1 veritas dcg inc. For each iteration a discrete fourier transform is performed. To improve spectral resolution of irregular sampling signals, this paper presents an antileakage nonuniform discrete fourier transform ndft. In the antileakage fourier transform, we first solve for the most energetic fourier coefficient, assuming that it causes the most severe leakage. Chapter 1 the fourier transform institute for mathematics. Mar 23, 2016 dal dominio del tempo a quello delle frequenze.
In the antileakage fourier transform, we first solve for the most energetic. The antileakage fourier transform alft is a regularization method using an iterative procedure for computing the spectrum of irregularly sampled data. The antileakage fourier transform alft is a fourier regularization method. These algorithms are established mainly on the basis of fourier transform. Pdf seismic data regularization, which spatially transforms irregularly sampled. Matching pursuit using a fourier basis is not a new method. Excessive highfrequency leakage causes ringing and can accentuate the. Half range fourier sine or cosine series a half range fourier sine or cosine series is a series in which only sine terms or. Us20090231956a1 method for interpolating seismic data by. Spectral leakage applies to all forms of dft, including the fft. This brings new challenges to seismic data regularization algo. Pdf antileakage fourier transform for seismic data.
And the properties of the discrete fourier transform, as well see, the differences in the properties between this and the fourier transforms weve talked about in previous lectures are related very closely to the implied periodicity in the sequence or in the discrete fourier. Fourier transform has many applications in physics and engineering such as analysis of lti systems, radar, astronomy, signal processing etc. Antileakage fourier transform for seismic data regularization in higher dimensions sheng xu1, yu zhang1, and gilles lambare2 abstract wideazimuth seismic data sets are generally acquired more sparsely than narrowazimuth seismic data sets. D032 seismic data regularization with anti leakage fourier transform sheng xu and don pham abstract 1 veritas dcg inc. An acceleration method for the antileakage parabolic radon.
In practice, the measured signals are limited in time and the fft calculates the frequency transform over a certain number of discrete frequencies called bins. Antileakage ndft and its application in data reconstruction ieee. Antileakage fourier transform among fourier based seismic interpolation methods, alft is an iterative algorithm that acts on each timefrequency slice separately, and reconstructs a temporal frequency spectrum along spatial dimensions. Antileakage leastsquares spectral analysis for data. This is a good point to illustrate a property of transform pairs.
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