Local similarity with the envelope as a seismic phase detector Sergey Fomel, The University of Texas at Austin, and Mirko van der Baan, University of Alberta SUMMARY We propose a new seismic attribute, local similarity with the envelope, as an indicator of localized phase of seismic signals. The proposed attribute appears to have a higher dynamical range and a better stability than the previously used local varimax norm. Synthetic and real data examples demonstrate the effectiveness of local similarity with the envelope in detecting and correcting time-varying, locally-observed phase of seismic signals. In this paper, we revisit the problem of phase estimation and propose a novel attribute, local similarity with the envelope, as a phase detector. Analogous to local kurtosis, local similarity with the envelope is defined using local similarity measurements via regularized least squares. We hypothesize that this attribute is maximized when the locally observed phase is close to zero. Advantages of the new attribute are a higher dynamical range and a better stability, which make it convenient for implementing automatic phase corrections. Using synthetic and field-data examples, we demonstrate properties and applications of the new attribute. INTRODUCTION Wavelet phase is an important characteristic of seismic signals. Physical causal systems exhibit minimum-phase behavior (Robinson and Treitel, 2000). For purposes of seismic interpretation, it is convenient to deal with zero-phase wavelets with minimum or maximum amplitudes centered at the horizons of interest. Zero-phase correction is a routine procedure applied to seismic images before they are passed to the interpreter (Brown, 1999). It is important to make a distinction between phases of propagating and locally observed wavelets. The former is the physical wavelet that propagates through the Earth, thereby sampling the local geology. It is subject to geometric spreading, attenuation, and concomitant dispersion. The latter is the wavelet as observed at a certain point in space and time. Its immediate shape results from its interaction (convolution) with the reflectivity of the Earth and the current shape of the propagating wavelet (Van der Baan et al., 2010). For instance, a thin layer with an outstanding seismic impedance has opposite polarities of seismic reflectivities at the top and the bottom interface, which make it act like a derivative filter and produce a wavelet with the locally observed phase subjected to a 90 rotation (Zeng and Backus, 2005). In the absence of well log information, it is often difficult to separate unambiguously the locally observed phase from the phase of the propagating wavelet. However, measuring the local phase can provide a useful attribute for analyzing seismic data (Van der Baan and Fomel, 2009; Van der Baan et al., 2010). Levy and Oldenburg (1987) proposed a method of phase detection based on maximization of the varimax norm or kurtosis as an objective measure of zero-phaseness. By rotating the phase and measuring the kurtosis of seismic signals, one can detect the phase rotations necessary for zero-phase correction (Van der Baan, 2008). Van der Baan and Fomel (2009) applied local kurtosis, a smoothly nonstationary measure (Fomel et al., 2007), and demonstrated its advantages in measuring phase variations as compared with kurtosis measurements in sliding windows. Local kurtosis is an example of a local attribute (Fomel, 2007a) defined by utilizing regularized leastsquares inversion. LOCALIZED PHASE ESTIMATION Figure 1 illustrates the fundamental difference between the propagating wavelet and the locally observed one. In this synthetic example, borrowed from Sengbush et al. (1961), a zerophase seismic wavelet is convolved with a reflectivity series to form a synthetic seismic trace. While isolated jumps in acoustic impedance produce isolated spikes in reflectivity, thin layers create a reflectivity series with its own phase and can rotate the apparent phase of the seismic trace by as much as 90. Figure 1: Synthetic impedance log (top) and corresponding seismic trace (bottom). The seismic trace, formed by convolving reflectivity with a zero-phase Ricker wavelet, exhibits a variable phase: 0 for an isolated interface and 90 for thin layers, possibly with a tuning effect on the amplitudes. Reproduced from Sengbush et al. (1961). Our goal is to estimate the time-variant, localized phase from seismic data. What objective measure can indicate that a certain signal has a zero phase? One classic measure is the varimax norm or kurtosis (Levy and Oldenburg, 1987). Varimax
is defined as N s 4 i φ = ( ) 2, (1) s 2 i where s i represents seismic amplitudes inside a window of size N. Varimax is simply related to kurtosis of zero-mean signals. Noticing that the correlation coefficient of two sequences a i and b i is defined as a i b i c[a,b] = N a 2 i b 2 i and the correlation of a i with a constant is a i (2) c[a,1] =, (3) N a 2 i Fomel et al. (2007) interpreted the varimax measure in equation (1) as the inverse of the squared correlation coefficient between s 2 i and a constant, φ = 1/c[s 2,1] 2. Well-focused or zero-phase signals exhibit low correlation with a constant and correspondingly high varimax (Figure 2). Figure 3: (a) 0 -phase Ricker wavelet compared with its envelope. (b) 90 -phase Ricker wavelet compared with its envelope. The 0 -phase has a higher correlation with the envelope. envelope picks the original signal. This example shows that, although kurtosis might be better at picking the true signal phase of the propagating wavelet, correlation with the envelope tends to pick better focused signals. more Additionally, as demonstrated in Figure 5, correlation with the envelope has a much higher dynamical range, which makes picking the optimal phase rotation easier. ENVELOPE SIMILARITY AS A LOCAL ATTRIBUTE The method of local attributes (Fomel, 2007a) is a technique of extending stationary or instantaneous attributes to smoothly varying or nonstationary attributes by employing a regularized least-squares formulation. In particular, one number for the correlation coefficient c in equation (2) is replaced with a set of numbers, c i, defined as a componentwise product of vectors c 1 and c 2, where c 1 = [ λ 2 I + S ( A T A λ 2 I )] 1 SA T b, (4) c 2 = [ λ 2 I + S ( B T B λ 2 I )] 1 SB T a. (5) Figure 2: (a) Squared 0 -phase Ricker wavelet compared with a constant. (b) Squared 90 -phase Ricker wavelet compared with a constant. The 90 -phase signal has a higher correlation with a constant and correspondingly a lower kurtosis. In this paper, we suggest a different measure, namely correlation between a signal and its envelope, for measuring the apparent phase of seismic signals. Zero-phase signals tend to exhibit higher correlation with the envelope (Figure 3). The two measures do not necessarily agree with one another, which is illustrated in Figures 4 and 5. For an isolated spike convolved with a compact wavelet, the two measures agree in the picking of the zero-phase result as having both a high kurtosis and a high correlation with the envelope (Figure 4). For a slightly more complex case of a double spike convolved with the same wavelet (Figure 5), the two measures disagree: kurtosis picks a signal rotated by 90 whereas correlation with the In equations (4-5), a and b are vectors composed of a i and b i, respectively; A and B are diagonal matrices composed of the same elements; and S is a smoothing operator. Regularized inversion appearing in equations (4-5) is justified in the theory of shaping regularization (Fomel, 2007b). The corresponding local similarity attribute has been used to align multicomponent and time-lapse images (Fomel, 2007a; Fomel and Jin, 2009), to detect focusing of diffractions (Fomel et al., 2007), to enhance stacking (Liu et al., 2009a,c), to create time-frequency distributions (Liu et al., 2009b), and to perform zero-phase correction with local kurtosis (Van der Baan and Fomel, 2009). In this paper, we apply it to zero-phase correction using local similarity with the envelope. We illustrate the proposed zero-phase correction procedure in Figure 6. The input synthetic trace contains a set of Ricker wavelets with a gradually variable phase (Figure 6(a)). We start with a number of phase rotations with different angles, each time computing the local inverse similarity with the envelope. The result of this step is displayed in Figure 6(b) and shows a clear high-similarity trend. After picking the trend, adding 90 to it, and performing the corresponding nonstationary trace rotation, we end up with the phase-corrected trace, shown in Figure 6(c). All original phase rotations are clearly detected and removed.
Figure 4: (a) Ricker wavelet rotated through different phases. (b) Correlation with the envelope (solid line) and varimax (dashed line) as functions of the phase rotation angle. (c) Inverse correlation with envelope (solid line) and inverse varimax (dashed line) as functions of the phase rotation angle. The two measures agree in picking the signal at 0 and 180. Note higher dynamical range of envelope similarity. Figure 5: (a) Ricker wavelet convolved with a double impulse and rotated through different phases. (b) Correlation with envelope (solid line) and varimax (dashed line) as functions of the phase rotation angle. (c) Inverse correlation with envelope (solid line) and inverse varimax (dashed line) as functions of the phase rotation angle. The two measures disagree by 90 in picking the optimal signal. The envelope similarity picks a better focused signal. Figure 6: (a) Input synthetic trace with variable-phase events. (b) Inverse local similar with the envelope as a function of the phase rotation angle. Red colors correspond to high inverse similarity. (c) Synthetic trace after non-stationary rotation to zero phase.
Figure 7: Input data: a section of the Boonsville dataset Figure 9: Nonstationary phase rotation picked from the similarity analysis shown in Figure 8. the result displayed in Figure 10. A zoomed-in comparison shows the effects of non-stationary phase correction: rotating major seismic events to zero degrees and improving their continuity. These effects can be useful both for improving seismic interpretation and for matching seismic data with well logs. Figure 8: Inverse local similar with the envelope as a function of the phase rotation angle, with application to the section from Figure 7. Red colors correspond to high inverse similarity. APPLICATION EXAMPLE An input dataset for our field-data example is shown in Figure 7 and represents a stacked section from the Boonsville dataset (Hardage et al., 1996). A zero-phase correction has been applied but has left regions with variable localized phase. The processing sequence is similar to the one applied in Figure 6. First, we apply a number of phase rotations with different angles and compute local similarity with the envelope after each rotation. The result is displayed in Figure 8. Next, we apply automatic picking with the algorithm described by Fomel (2009) to extract the nonstationary phase rotation that maximizes the envelope similarity. The result is displayed in Figure 9. Finally, the phase correction is applied to the data, with Figure 10: Zoomed-in comparison of the data before phase correction (a) and after phase correction (b). Nonstationary phase correction helps in identifying significant horizons and increasing their resolution in time. CONCLUSIONS We have presented an approach to nonstationary apparent phase identification, which is based on a novel attribute: local similarity with the envelope. In synthetic and field-data examples, local similarity with the envelope exhibits a higher dynamical range than the previously used local kurtosis and a tendency to pick focused signals. Practical applications of the envelope similarity for zero-phase correction of seismic signals should be combined with well log analysis in order to separate the locally-observed phase from the propagating-wavelet phase.
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