M.Sc. Student of Geophysics, Geology Department, Faculty of Science, Urmia University, Iran
Professor, Earth Physics Department, Institute of Geophysics, University of Tehran, Iran
Assistant Professor, Geology Department, Faculty of Science, Urmia University, Iran
*نگارنده رابط: تلفن: 61118277-021 دورنگار: 88630479-021 E-mail: email@example.com
Discrete Fourier Transform (DFT) is the basic part of various algorithms of signal processing in many fields of science and technology. 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. Consequently, spectrum leakage appears in frequency domain. Energy leakage of DFT spectrum can also occur due to non-uniform sampling in time or spatial domain and usually is more serious. The leakage can hide smaller spikes among actual spectrum and is an important factor that affects spectrum estimation. In practice, we should try to reduce the energy leakage of DFT spectrum to improve the resolution of frequency spectrum. For evenly sampled signals, suppressing approaches of spectrum leakage are diverse; most common method among them is windowing method. Study on how to suppress spectrum leakage of non-uniform Fourier transform is important both in theory and practice. Here we introduce and apply an anti leakage Fourier transform (ALFT) algorithms for suppressing spectrum leakage of non-uniform Fourier transform, improving the resolution of temporal frequency spectrum or spatial wave number spectrum.
One of the areas of study that has the same problem is seismic exploration technology. Seismic data sets are generally irregularly sampled in inline midpoints, cross-line midpoints, offset and azimuth. This irregular sampling can limit the effectiveness of high end 3D de-multiple and imaging algorithms such as 3D surface related multiple elimination, wave equation pre-stack depth migration and many other processes. To overcome this issue, it is common in seismic data processing to use regularization and interpolation. Interpolation processes fill the missing traces and regularization transfer traces from their irregular recorded location to locations on a regular grid. We apply ALFT for seismic data interpolation and regularization that leads to reconstruction of seismic data on a regular grid.
ALFT is an iterative algorithm that acts on frequency slices and reconstruct each temporal frequency spectrum along spatial dimensions. For an input data with N_p known samples, the original algorithm of ALFT can be performed as follow:
1- Computing Fourier components of the data using equation 1.
2- Selecting the largest coefficient and adding it to the precomputed coefficients.
3- Updating data by subtracting the contribution of selected coefficient (equation 2) from input data (equation 3).
4- Iterating steps 2 and 3 until reaching the threshold.
The idea of ALFT is simple and intuitive: first seismic data will be transformed to f -x domain, by applying DFT the f -k spectrum of data will be estimated. The largest Fourier coefficient is selected and subtracted from the input data. In the subsequent iterations, successive maximum components are subtracted until the norm of the residual is negligible. This iterative processes is able to recover a sparse spectrum that, when evaluated at sampling points, approximates regularly sampled data. This method relies on the common assumption that sparsely sampled data can be represented by a few Fourier components. ALFT can handle pure non-uniform seismic data and uniform seismic data with gap and missing traces. For regular data sets, by applying an anti-alias mask ALFT can handle steep dips. Generalization of ALFT to higher dimensions is simple and straightforward and for high dimension data ALFT can reconstruct very sparse data sets. Performance of the method was tested on both synthetic and real seismic data.
We applied ALFT algorithm to reconstruction of synthetic and real seismic data sets. The results show the effectiveness of ALFT in interpolation and regularization of input data on any desired regular sampling grid. Compared to those interpolation methods that use FFT, ALFT has a slow procedure. However, computing the DFT’s in small windows of data sets, greatly reduces the computational cost of the algorithm. On the other hand, when the input data sets are sampled on a regular grid which has missing traces or gaps, one can use FFT instead of DFT to compute Fourier transform. ALFT reconstruction method suffers much less from edge effects and gibs phenomenon.
The sequence of computing Fourier coefficients from maximum energy to minimum energy, and subtraction of contribution of them from remained data, plays a key rule in ALFT algorithm.