Interpolation of nonstationary seismic records using a fast non-redundant S-transform
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1 Interpolation of nonstationary seismic records using a fast non-redundant S-transform Mostafa Naghizadeh and Kris A. Innanen CREWES University of Calgary SEG annual meeting San Antonio 22 September 2011
2 Outline 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
3 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
4 Goal Using a fast and non-redundant transform for interpolation of curved seismic events without spatial windowing of data.
5 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
6 Introduction I Fourier Transform Stationary signals: constant frequency/wavenumber content at all time/space samples. Time-frequency analysis Nonstationary signals: Dynamic frequency/wavenumber content. Methods for analysis of nonstationary signals: Short-time Fourier transform (i.e. Gabor transform) Wavelet transform (Progressive resolution) S-transform (Frequency dependent windowing) Curvelet transform (Local and directional decomposition) Remember: Fourier transform is the core building block for all time-frequency analyses.
7 Introduction II Nonstaionary signals in seismic data processing Time axis: Dispersive and attenuated seismic traces. Spatial axis: Hyperbolic and parabolic events Dispersive events Discontinuities Interpolation of spatially nonstationary seismic data Spatial windowing before interpolation Adaptive prediction filters [Naghizadeh and Sacchi, 2009] Curvelet interpolation [Naghizadeh and Sacchi, 2010]
8 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
9 The main source [Brown et al., 2010] The source
10 S-transform in time domain [Stockwell et al., 1996] S-Transform (Stockwell et al., 1996) S-Transform (Time lag, Frequency) Signal Frequency-dependent time window Fourier Kernel
11 S-transform in frequency domain [Box function (I)] FFT Shifting Multiply 1D IFFT
12 Box function (II) Amplitude Time samples Frequency shift Amplitude Frequency shift Frequency shift Time samples
13 Box function (III) Amplitude Time samples Frequency shift Amplitude Frequency shift Frequency shift Time samples
14 Box function (IV) Amplitude Time samples Frequency shift Amplitude Frequency shift Frequency shift Time samples
15 Chirp function Amplitude Time samples Frequency shift Amplitude Frequency shift Frequency shift Time samples
16 Sine function Amplitude Time samples Frequency shift Amplitude Frequency shift Frequency shift Time samples
17 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
18 A second look at chirp function
19 Dyadic sampling of Alpha-domain
20 In-place non-redundant S-Transform f
21 In-place non-redundant S-Transform Fast generalized Fourier transform (FGFT) [Brown et al., 2010] t Generalized Fourier Transform (FGFT)) t f FFTs t interpretation f
22 FGFT of a box function Amplitude Amplitude Time samples FNST samples Amplitude 1 0 Time samples
23 FGFT of a chirp function Amplitude Amplitude Time samples FNST samples Amplitude Time samples
24 FGFT of a hybrid chirp function Amplitude Amplitude Time samples FNST samples Amplitude Time samples
25 FGFT of a sine function Amplitude Amplitude Time samples FNST samples Amplitude Time samples
26 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
27 Regularized least-squares FGFT interpolation The FGFT interpolation can be achieved by minimizing the following cost function [Tikhonov and Goncharsky, 1987] d: Observed data in t-x domain g: FGFT coefficients T: Sampling function G: FGFT operator W: Weight function µ: Trade-off parameter J = d T G H Wg µ 2 g 2 2. The weight function W needs to be properly identified from the data for optimal recovery of missing samples.
28 Derivation of weight function W Irregularly missing samples For irregularly missing samples a weight function which renders a sparse representation of FGFT coefficients is desired [Naghizadeh and Innanen, 2011] Initialization W 0 = I For k = 1, 2, 3... ĝ k = argmin{ d obs T G T W k 1 g µ2 g 2 2 } g W k = diag(abs(ĝ k )) End Regularly missing samples For regularly missing samples a priori information is needed to preserve specific FGFT coefficients. In the case of seismic data this can be extracted from low frequencies. [Spitz, 1991]
29 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
30 Reconstruction of a chirp function Amplitude Time samples Amplitude FNST samples Amplitude Time samples
31 Irregularly sampled chirp function Amplitude Amplitude Time samples
32 Regularly sampled chirp function (without a priori information) Amplitude Amplitude Time samples
33 Regularly sampled chirp function (with a priori information) Amplitude Amplitude Time samples
34 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
35 FGFT interpolation for seismic data Irregularly missing traces 1 Transform the original data to f-x domain. 2 Apply least-squares FGFT interpolation to each frequency. 3 Transform back the results of step 2 to t-x domain. Regularly missing traces 1 Transform the original data to f-x domain. 2 Compute FGFT of half frequencies. 3 Create a mask function by thresholding (1 for elements above threshold value and 0 otherwise)the FGFTs of step 2. 4 Interlace zero traces between each trace of step 1. 5 Apply least-squares FGFT interpolation to each frequency of step 4 using the mask function from step 3. 6 Transform back the results of step 5 to t-x domain.
36 Original data with hyperbolic events Original data with hyperbolic events
37 Decimated data Decimated data
38 FGFT interpolated data Reconstructed data
39 Original data Original data with hyperbolic events
40 Difference section Difference section
41 f-k spectra of data F-K domain representation of data Original Decimated Reconstructed
42 Seismic section with conflicting dip events a) Distance (m) b) Distance (m) c) Distance (m) Time (s) 0.7 Time (s) 0.7 Time (s) d) Normalized wavenumber Kx e) Normalized wavenumber Kx f) Normalized wavenumber Kx
43 Irregularly sampled seismic section a) Offset (m) b) Offset (m) c) Offset (m) Time (s) 0.7 Time (s) 0.7 Time (s) d) Normalized wavenumber Kx e) Normalized wavenumber Kx f) Normalized wavenumber Kx
44 1 Introduction Motivation Overview 2 Fourier analysis of nonstationary signals S-transform in frequency domain 3 Fast generalized Fourier transform (FGFT) A non-redundant fast S-transform 4 Fast generalized Fourier interpolation Regularized least-squares optimization 1D synthetic examples Synthetic seismic data Real seismic data 5 Conclusions
45 A near offset section from Gulf of Mexico Distance (m) Time (s)
46 FGFT interpolated data Distance (m) Time (s)
47 Adaptive f-x interpolation Distance (m) Time (s)
48 Curvelet interpolation Distance (m) Time (s)
49 Conclusions Fast Generalized Fourier Transform (FGFT) is a fast and efficient way for analyzing nonstationry signals. FGFT does not increase the size of data (non-redundant) and is very simple to apply. The fast and non-redundant properties of FGFT comes with trade-off on precision which might make it unsuitable for some applications. FGFT can be used to interpolate: Irregular sampling: Combination of a nonstationry signal and irregular sampling is hard to be reconstructed. FGFT shows somehow stable recovery in this situation. Regular sampling: Regular sampling creates high amplitude artifact in Fourier domain. In FGFT, these artifact are only present at high frequencies. For a successful FGFT interpolation a priori mask function is required.
50 Conclusions (Cont.) For regularly sampled aliased nonstationary seismic data one can make a mask function from half frequencies in the f-x domain of original data. The mask function can be used to guide FGFT interpolation. The extension of FGFT to the multidimensional case is straightforward. Another seismic application for FGFT SI P2 FWI Theory I Tuesday, September 20, 2011 Room: Ballroom C On the calibration of a fast S-transform with application to AVF inversion of anelastic reflectivity Chris Bird, K. A. Innanen, L. R. Lines, and M. Naghizadeh
51 Acknowledgments Sponsors of CREWES at the University of Calgary. Dr. Robert A. Brown Dr. Mauricio D. Sacchi Dr. Sam T. Kaplan
52 Brown, R. A., M. L. Lauzon, and R. Frayne, 2010, A general description of linear time-frequency transforms and formulation of a fast, invertible transform that samples the continuous s-transform spectrum nonredundantly: IEEE Trans. Signal Processing, 58, Naghizadeh, M. and K. A. Innanen, 2011, Seismic data interpolation using a fast generalized fourier transform: Geophysics, 76, V1 V10. Naghizadeh, M. and M. D. Sacchi, 2009, f-x adaptive seismic-trace interpolation: Geophysics, 74, V9 V , Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data: Geophysics, 75, WB189 WB202. Spitz, S., 1991, Seismic trace interpolation in the F-X domain: Geophysics, 56, Stockwell, R. G., L. Mansinha, and R. P. Lowe, 1996, Localization of the complex spectrum: The s transform: IEEE Trans. Signal Processing, 44, Tikhonov, A. N. and A. V. Goncharsky, 1987, Ill-posed problems in the natural sciences: MIR Publisher.
53 Extending FGFT to higher dimensions: t-x domain Distance (m) Time (s)
54 Extending FGFT to higher dimensions: f-k domain
55 Extending FGFT to higher dimensions: 2D FGFT domain
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