GEOPHYSICS, VOL. 73, NO. 1 共JANUARY-FEBRUARY 2008兲; P. V1–V9, 8 FIGS. 10.1190/1.2795396
Wavelet-based detection of singularities in acoustic impedances from surface seismic reflection data
Chun-Feng Li1 and Christopher Liner2
N is the vanishing moment of the seismic wavelet. This theoretical approach forms the basis of linking singularity exponents 共Hölder exponents兲 in acoustic impedance with those computable from seismic data. By using wavelet-based multiscale analysis with complex Morlet wavelets, we can estimate singularity strengths and localities in subsurface impedance directly from surface seismic data. Our results indicate that rich singularity information in acoustic impedance variations can be preserved by surface seismic data despite data-acquisition and processing activities. We also show that high-resolution detection of singularities from real surface seismic data can be achieved with a proper choice of the scale of the mother wavelet in the wavelet transform. Singularity detection from surface seismic data thus can play a key role in stratigraphic analysis and acoustic impedance inversion.
ABSTRACT Although the passage of singularity information from acoustic impedance to seismic traces is now well understood, it remains unanswered how routine seismic processing, mode conversions, and multiple reflections can affect the singularity analysis of surface seismic data. We make theoretical investigations on the transition of singularity behaviors from acoustic impedances to surface seismic data. We also perform numerical, wavelet-based singularity analysis on an elastic synthetic data set that is processed through routine seismic processing steps 共such as stacking and migration兲 and that contains mode conversions, multiple reflections, and other wave-equation effects. Theoretically, seismic traces can be approximated as proportional to a smoothed version of the 共N Ⳮ 1兲th derivative of acoustic impedance,where
interpretation. Because seismic reflections normally occur at sharp impedance boundaries, mostly related to faults and lithologic interfaces, singularity attributes can help in seismic structural and stratigraphic interpretation. Since the mid-1990s, wavelet transforms have been used widely in seismic data processing, but they have only recently been applied to data interpretation through extraction of singularity information. Herrmann and coworkers 共Herrmann and Stark, 2000; Lyons and Herrmann, 2001; Herrmann, 2001兲 are among the first to detect isolated singularities from seismic data by using both multiscale and monoscale techniques. Li 共2002, 2004兲 and Li and Liner 共2005兲 apply a multiscale technique for pointwise detection of seismic singularities and recognize the Hölder exponent as a useful 3D seismic attribute with important new properties. In particular, they discover that acoustic impedance singularity information can be detected directly from surface seismic reflection data.
INTRODUCTION Surface seismic reflection amplitudes originate from acoustic impedance variations. Phase and frequency changes are more subtle and difficult to detect and therefore have less influence on seismic interpretation than amplitude. But it is the combination — amplitude, phase, and frequency — that should be used equally and jointly to extract the maximum information from seismic data. The continuous wavelet transform 共CWT兲 leads to a new seismic attribute 共Herrmann and Stark, 2000; Lyons and Herrmann, 2001; Li, 2002, 2004; Liner et al., 2004; Smythe et al., 2004; Li and Liner, 2005兲 termed the Hölder exponent, which is not dependent on waveform amplitude, frequency, or phase alone but on the combination of the three. The Hölder exponent, or Lipschitz exponent 共Mallat and Hwang, 1992兲, measures the singularity of a point or a small confined region in a data set. Singularities in seismic and acoustic impedance data carry important information that has not been used fully in seismic
Manuscript received by the Editor 4 March 2007; revised manuscript received 4 August 2007; published online 9 November 2007. 1 Tongji University, State Laboratory of Marine Geology, Shanghai, China. E-mail:
[email protected]. 2 Saudi Aramco, EXPEC Advanced Research Center, Dhahran, Saudi Arabia. E-mail:
[email protected]. © 2008 Society of Exploration Geophysicists. All rights reserved.
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Li 共2002, 2004兲 and Li and Liner 共2005兲 test the singularity algorithm on synthetic and field data. The physical basis of the synthetic test was the 1D convolutional model, which generates seismograms devoid of multiples, mode conversions, and attenuation effects. Liner et al. 共2004兲 and Smythe et al. 共2004兲 apply the algorithm to field data and tie results to electric and gamma well logs but do not analyze the relationship between singularity in seismic and log impedances. In this study, we first investigate theoretically how singularity information in acoustic impedances can be transmitted to surface seismic data. We then use both simulated and real seismic data to determine if and how we can detect singularities in acoustic impedance directly from surface seismic data. We will test whether high-resolution singularity information is obtainable. The present study uses synthetic data from the Marmousi2 model 共Martin, 2004兲 generated by finite-difference modeling of the elastic wave equation and then imaged through a full seismic processing flow. This approach allows us, for the first time, to relate singularity behavior estimated from elastic synthetic seismic data to characteristics of the impedance itself.
THEORETICAL CONSIDERATIONS Real well-logging data show that acoustic impedance transitions 共T兲 are not characterized only by simple step functions 共Herrmann, 1997, 1998兲. We represent the spectrum of transition behavior by using generalized onset functions 共Gelfand and Shilov, 1964; Holschneider, 1995; Herrmann, 1997, 1998兲, in the form
TⳭ共z, ␣ 兲 ⳱
冦
0
zⱕ0
兩z兩␣ z⬎0 ⌫ 共␣ Ⳮ 1兲
In the sense of a generalized function, reflectivity R共z兲 often can be related to acoustic impedance I共z兲 共Peterson et al., 1955; Walden and Hosken, 1985; Todoeschuck and Jensen, 1988兲 through
R共z兲 ⬇
S共z兲 ⬇
Tⳮ共z, ␣ 兲 ⳱
冦
冧
共1兲
共2兲
where v0 and z0 are reference velocity and depth, respectively. Note that equation 2 is a parameterization nearly identical to the generalized transition in equation 1. Wapenaar 共1998, 1999兲 consequently finds that the normal-incidence reflection and transmission coefficients are dependent on frequency and ␣ , and that in the seismic frequency range, smoothing on the singular interface has hardly any effect on seismic response. The 1D convolutional model in the depth domain states that a surface seismogram S共z兲 is the convolution of the seismic wavelet s共z兲 and the reflectivity R共z兲,
S共z兲 ⳱ R共z兲 ⴱ s共z兲.
共5兲
The logarithmic operator on I共z兲 certainly can alter the singularity information in I共z兲 by weakly smoothing I共z兲 and therefore slightly increasing the Hölder exponent. However, if I共z兲 has a relatively small variance locally, which can be the case in the depth range of petroleum seismology, the logarithm of impedance can be linearly approximated locally by
ln I共z兲 ⳱ CII共z兲,
共6兲
where CI is a constant and subscript I indicates impedance. In other words, logarithmic operations on I共z兲 will not significantly alter the local character of I共z兲 within the compact region of the analyzing wavelet, and thus the local singularity behavior is preserved. With this approximation, equation 5 becomes
S共z兲 ⬇ C
d I共z兲 ⴱ s共z兲, dz
s共z兲 ⳱
where z is the depth, ␣ the Hölder exponent, and ⌫ the Gamma function. Wolf 共1937兲 studied the reflection of elastic waves from a transition layer whose velocity function is represented by a linear ramp. Wapenaar 共1998, 1999兲 made a generalization to Wolf’s work and investigated the theory of seismic reflection and transmission coefficients by using a velocity function
v共z兲 ⳱ v0共兩z/z0兩兲␣ ,
1 d ln I共z兲 ⴱ s共z兲. 2 dz
共7兲
where C ⳱ CI /2. In the depth domain, by supposing that the seismic wavelet has N vanishing moments, we can write 共Mallat and Hwang, 1992; Staal, 1995兲
冧
兩z兩␣ zⱕ0 ⌫ 共␣ Ⳮ 1兲 , 0 z⬎0
共4兲
By substituting equation 4 into equation 3, we have
and
ⳮ
1 d ln I共z兲. 2 dz
共3兲
dN 共z兲, dzN
共8兲
where 共z兲 is a smoothing function, which is the integral of a wavelet with one vanishing moment. By using equation 8 and the derivative theorem of convolution 共Bracewell, 1965兲, we now can further cast equation 7 into
S共z兲 ⬇ C
d dNⳭ1 dN I共z兲 ⴱ N 共z兲 ⳱ C NⳭ1 关I共z兲 ⴱ 共z兲兴 dz dz dz
⳱C
dNⳭ1 I共z兲 ⴱ 共z兲. dzNⳭ1
共9兲
Smoothing by 共z兲 on local singularities can be viewed mathematically as taking fractional integration of order p for small-scale singularities. For example, for a general transition TⳭ in the form of equation 1, the fractional integration of order p on TⳭ can be written as 共Gelfand and Shilov, 1964; Holschneider, 1995兲
冋
dⳮp z␣ dⳮp 关T 共z ⬎ 0, ␣ 兲兴 ⳱ Ⳮ dzⳮp dzⳮp ⌫ 共1 Ⳮ ␣ 兲 ⳱ ⳱
1 ⌫ 共p兲⌫ 共1 Ⳮ ␣ 兲
冕
z
册
t␣ 共z ⳮ t兲 pⳮ1dt
0
1 z␣ Ⳮp , ⌫ 共1 Ⳮ p Ⳮ ␣ 兲
共10兲
Wavelet-based detection of singularity where p ⬎ 0, z ⬎ 0, and ␣ ⬎ⳮ1. In classical calculus, ␣ ⬎ⳮ1 is assumed for the integral in the left-hand side to be convergent. In the context of generalized functions, this assumption can be relaxed 共Gelfand and Shilov, 1964兲. It is clear from equation 10 that fractional integration of order p increases the original Hölder exponent in the transition function from ␣ to ␣Ⳮp. Therefore, equation 9 can become
冋
dNⳭ1 dNⳭ1 dⳮp S共z兲 ⬇ C NⳭ1 关I共z兲 ⴱ 共z兲兴 ⳱ C NⳭ1 I共z兲 dz dz dzⳮp ⳱C
冋
册
dⳮp dNⳭ1 dNⳭ1ⳮp I共z兲 ⳱ C I共z兲, dzⳮp dzNⳭ1 dzNⳭ1ⳮp
册
共11兲
where p is dependent on the degree of smoothing played by 共z兲. Equations 9 and 11 suggest that seismic traces can be taken as proportional to a smoothed version of the 共N Ⳮ 1兲th derivative of acoustic impedance. The global scaling exponent H is a measure of the overall differentiability or complexity of a data series and can be calculated in the frequency domain 共Li, 2003兲. The local counterpart of H is the local Hölder exponent, which is measurable in the wavelet domain. If the global scaling exponent of I共z兲 is HI, the global scaling exponent of S共z兲 becomes HS 共Li, 2004兲 ,
HS ⳱ HI ⳮ 共N Ⳮ 1 ⳮ p兲 ⳱ HI ⳮ N ⳮ 1 Ⳮ p, 共12兲 where subscript S indicates seismic data. Locally, if I共z兲 is to be parameterized by equations 1 and 2 with local Hölder exponents ␣ I, the local Hölder exponents of S共z兲 apparently will approach
␣ S ⳱ ␣ I ⳮ 共N Ⳮ 1 ⳮ p兲 ⳱ ␣ I ⳮ N ⳮ 1 Ⳮ p. 共13兲
共14兲
WAVELET-BASED MULTISCALE ANALYSIS From equation 11, the wavelet transform of S共z兲 becomes
冋
WT关S共z兲兴 ⬇ CWT
册
dNⳭ1ⳮp I共z兲 . dzNⳭ1ⳮp 共15兲
Equation 15 shows that the wavelet transform of surface seismic data is approximately proportional to the wavelet transform of the 共N Ⳮ 1 ⳮ p兲th derivative of acoustic impedance. Theorems for estimating the Hölder exponent from wavelettransform coefficients are well established 共Holschneider and Tchamitchian, 1989; Jaffard, 1989; Mallat and Hwang, 1992兲. In the cone of influ-
ence along each modulus-maxima line, the theorem takes the form of
兩WT关f共z兲兴兩 ⬇ A ␣ f for → 0,
共16兲
in which is the scaling factor and A is a constant. Thus, equation 15 can be understood as
A1 ␣ S ⬇ A2 ␣ IⳮNⳮ1Ⳮp ,
共17兲
where the exponents follow from equation 13. Singular points in I共z兲 and S共z兲 are packed closely, and singularity information can be buried deeply in the seismic phase. To better deal with this issue, complex wavelets are preferred over real ones. We use complex Morlet wavelets to calculate Hölder exponents from seismic data. For complex Morlet wavelets, the CWT coefficients are also complex numbers. However, the Hölder exponent is a real number estimated from the moduli of the wavelet-transform coefficients, as indicated by equation 16. In connection to time-series analysis using this wavelet, it has been noted that modulus-maxima lines in time-scale space tend to be linear directly above the singular points 共Seuret and Gilbert, 2000; Li, 2002, 2004兲. Thus, time-consuming searches for curved local modulus-maxima lines can be eliminated so that the Hölder exponent calculation can be simplified and the speed improved. These are important issues for singularity analysis of large 3D seismic volumes. We constructed a model with isolated singular points 共Figure 1a兲 to test the analyzing algorithm. In the model, there are five singular points, with known Hölder exponents ranging fromⳮ1 to 1. The analyzing technique faithfully recovered the Hölder exponents at those points 共Figure 1b兲. We notice that in the neighborhood of each singular point, this algorithm seems to produce artifacts in the cone of influence of the continuous wavelet transform. Such artifacts are not a problem for pointwise detection of singularities because the artifacts
4 3 2 1 0
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␣ S ⳱ ␣ I ⳮ 3 Ⳮ p.
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Although smoothing by 共z兲 appears to be troublesome, studies show that in the seismic frequency band, smoothing plays only a secondary role in altering Hölder exponents 共Wapenaar, 1998; Li, 2004兲. As an example, for a Ricker seismic wavelet with two vanishing moments, the local Hölder a) 7 exponents computed from seismic data can be re6 lated to the local Hölder exponents calculated 5 from impedance by
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Figure 1. 共a兲 A testing model with five singular points of different Hölder exponents; ␣ → 0 indicates that ␣ approaches 0 asymptotically. 共b兲 Estimated Hölder exponents using wavelet-based multiscale analysis.
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largely are suppressed when singular points are packed closely. This statement is true for acoustic impedance and seismic traces in which each data point is to be considered a singular point 共Li and Liner, 2005兲. Figure 2a shows the wavelet transform of the singular model in Figure 1a. Also shown in Figure 2 are Hölder exponent analyses at data point 1500 using both the modulus-maxima line 共Figure 2b and c兲 and a straight line directly above the singular point, as in our im-
a)
plementation 共Figure 2d and e兲. The black dots in Figure 2b are wavelet-transform modulus-maxima points 共WTMMP兲, with which an exact Hölder exponent can be calculated in Figure 2c from the theory of Mallat and Hwang 共1992兲. Because WTMMP in Figure 2b can be approximated well by a linear line in Figure 2d, it follows that both methods give very close estimates of Hölder exponents. Hölder exponents are computed from the slopes of the linear least-squares fits shown in Figure 2c and e, in the double-logarithmic space of scales versus squared moduli of CWT coefficients. Mathematically, this process amounts to 共Holschneider and Tchamitchian, 1989; Mallat and Hwang, 1992; Daubechies, 1992兲
log2兩XCWT兩2 ⱕ log2 C Ⳮ 共2␣ Ⳮ 1兲log2 , 共18兲 where ␣ is the Hölder exponent, is the scale, and C is a constant.
THE MARMOUSI2 MODEL AND SYNTHETIC DATA
b)
c)
d)
e)
Figure 2. 共a兲 Moduli of CWT coefficients of the testing model in Figure 1a. 共b兲 Moduli of CWT coefficients around point 1500 and the modulus-maxima points 共black dots兲. 共c兲 Estimation of Hölder exponent from the linear least-squares regression of the modulusmaxima points in the double-logarithmic space of scales versus squared moduli of CWT coefficients. 共d兲 Moduli of CWT coefficients around point 1500. 共e兲 Estimation of Hölder exponent from the linear least-squares regression using the points on the linear line directly above the singular point 共shown in d兲. In both 共c兲 and 共e兲, Hölder exponents are estimated by using equation 18 from the slope of the linear least-squares regressions.
The Marmousi2 model 共Figure 3兲 is a new generation of elastic, heterogeneous geologic model containing a variety of structural and stratigraphic features 共Martin, 2004兲. Broadband synthetic seismic data were acquired over this complex model using an elastic finite-difference scheme that is second-order accurate in time and fourthorder accurate in space 共Martin, 2004兲. With elastic wave-equation forward modeling, various events such as multiples, mode conversions, head waves, and all primaries are present. The data processing included geometry assignment, datum corrections, multiple attenuation, and prestack depth migration. This data set presents a unique opportunity for studying how singularity information in the earth model can be transferred to surface seismic data because 共1兲 such data include physical effects that are lacking in a 1D convolutional model and 共2兲 the data have undergone conventional data-processing procedures that make it possible to study whether processing can disrupt the passage of singularity information from earth model to data. We have both Kirchhoff and wave-equation prestack depth-migrated data sets 共Martin, 2004兲 available in this study 共Figures 4 and 5兲. We applied wavelet-based singularity analysis to both migrated images 共Figures 4 and 5兲. The Marmousi2 data set allows a direct comparison between the singularity pattern of the acoustic impedance model and the Hölder exponents computed from the seismic data. By comparing the seismic sections with the Hölder sections in Figures 4 and 5, we notice that seismic data do carry rich singularity information from acoustic impedance. Because Hölder exponents characterize the sharpness of geologic transitions, it is not surprising to note that Hölder attributes highlight the
Wavelet-based detection of singularity
Figure 3. A part of the Marmousi2 acoustic impedance model of Martin 共2004兲. The downward-pointing black arrow in the upper left corner indicates the location of the acoustic impedance trace shown in Figure 6. This trace, at the horizontal distance of 3.0 km in the Marmousi2 model, corresponds with the synthetic seismic trace at CDP 600.
Figure 4. 共a兲 Kirchhoff-migrated synthetic seismic section based on the Marmousi2 model 共Martin, 2004兲. Scale to right shows relative amplitude. 共b兲 Calculated Hölder attributes from 共a兲. Scale to right shows the singularity exponent, which has no units. The downwardpointing black arrows in the upper left corner indicate the location of the acoustic impedance trace shown in Figure 6.
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geologic boundaries 共faults and lithologic interfaces兲 in acoustic impedances. However, the detection of singularity information from surface seismic data is dependent on the seismic data quality and imaging schemes. Figures 4 and 5 indicate that wave-equation migration gives a better seismic image 共Figure 5a兲, particularly in the center part of the model, which leads to a better image of the Hölder attribute 共Figure 5b兲. The dense layered model imaged by the Hölder attribute is visually similar to impedance inversion results but is fundamentally different in meaning and method of computation. The singularity image is calculated directly from the migrated data traces alone, unlike impedance inversion that requires well-log calibration for accurate results. As mentioned earlier, the Hölder attribute is not determined solely by amplitude and cannot be related directly to the impedance contrast. The Hölder attribute is not a substitute for impedance but can aid in impedance inversion. Furthermore, the Hölder attribute is a single-channel process that operates on one trace at a time, meaning that
Figure 5. 共a兲 Wave-equation-migrated synthetic seismic section based on the Marmousi2 model 共Martin, 2004兲. Scale to right shows relative amplitude. 共b兲 Calculated Hölder attributes from 共a兲. Scale to right shows the singularity exponent, which has no units. The downward-pointing black arrows in the upper left corner indicate the location of the acoustic impedance trace shown in Figure 6.
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it applies equally well to 1D, 2D, or 3D migrated images. Smythe et al. 共2004兲 showed that in 3D, the vertical cross-sectional view of the Hölder attribute contains important information related to seismic stratigraphy, whereas the horizontal map view carries information similar to but more general than coherence images. Some of the key features in the Marmousi2 model 共Martin, 2004兲 also are shown on the Hölder singularity attribute section 共Figure 4b兲. The singularity attribute reveals sand channels of different types 共water-wet, gas-charged, or oil-charged channels兲, as well as faults and unconformities. The Hölder attribute reveals boundaries in the data and therefore aids both stratigraphic and structural interpretations. In general, lower Hölder exponents indicate higher singularity strengths and sharper transitions in seismic data and impedances, whereas higher exponents might correspond to places of blocky or uniform structures, such as the homogeneous salt structure shown in Figure 4b.
a)
Seismic traces at common-depth-point 共CDP兲 600 共Figure 6兲 have been extracted from the Kirchhoff and wave-equation prestack migration data, along with the acoustic impedance profile from the same location. Figure 6 is a comparison between acoustic impedance 共Figure 6c兲 and Hölder exponents from the synthetic traces 共Figure 6b and d兲. It is observed that calculated Hölder exponents from Kirchhoff-migrated data 共Figure 6a and b兲 are similar to those from wave-equation-migrated data in their overall shapes 共Figure 6d and e兲, although Kirchhoff-migrated data seem to be more singular and show smaller Hölder exponents. A comparison between the acoustic impedance and seismic traces indicates that seismic reflection data are characterized by high-frequency oscillations around zero amplitude, which makes it difficult to see singularities inherited from acoustic impedance. Hölder exponents calculated from synthetic seismic traces, on the other hand, ac-
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–1 –2 –3 500
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S(z)
0.1 0 –0.1 –0.2
[x 10 3 (m g)/(cm 3 s)]
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d) S(z)
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Figure 6. Hölder exponents calculated from synthetic seismic traces accurately detect the strengths and localities of singular transitions 共dashed lines兲 in the acoustic impedance. 共a兲 Estimated Hölder exponents from Kirchhoff-migrated synthetic seismic trace in 共b兲. 共b兲 Kirchhoff-migrated synthetic seismic trace at CDP 600. 共c兲 The acoustic impedances corresponding to the seismic trace at CDP 600. 共d兲 Wave-equation-migrated synthetic seismic trace at CDP 600. 共e兲 Estimated Hölder exponents from the wave-equation-migrated synthetic seismic trace in 共d兲.
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ure 7c and e. This difference is caused by the fact that unlike the theocurately detect the strengths and localities of singular transitions in the acoustic impedance 共Figure 6兲. retical model in Figure 1, seismic traces are essentially band limited, At several depths 共dashed lines兲 in Figure 6, we can observe that and constant-slope behavior can exist only for a finite scale range sharp transitions in acoustic impedance are detected by Hölder expo共Wapenaar, 1998兲. The choice of scale range is dependent on the nent minima. Site A in I共z兲 is a large step 共Figure 6c兲 associated with bandwidth of the seismic data. a swarm of seismic oscillations, but the associated Hölder exponent Linear regression on data points outside the frequency or scale pinpoints the step in I共z兲. Site B in I共z兲 is a minor step, but it also is range is prone to error. For more accurate computing, the ideal situawell detected. SitesAand B have quite different impedance contrasts tion would be to choose the scale range for each singular point. Howbut similar computed Hölder exponents. This observation verifies one important Hölder exponent a) property. It is a measure of transition character rather than impedance contrast across the boundary. As such, the Hölder exponent is complementary to seismic amplitude, which originates from the impedance contrast. At site C, two closely spaced steps mimic a sharp singular point that causes a low Hölder exponent. Note that direct inspection of the locality of this singular feature on seismic traces would be inconclusive. Sites D and E are blocky boundaries of a high-impedance zone, also clearly revealed by Hölder exponents. It seems that in the seismic traces, singular points are located near b) c) the takeoff points of sharp amplitude jumps. In particular, the stronger singularity at site E compared to that at site D is documented by the Hölder exponent. Site E is also slightly more singular than sites A and B, and the Hölder exponent calculated from the Kirchhoff image at site E is correspondingly smaller. These observations are in line with the theoretical considerations already presented and prove that singularity information in acoustic impedance can be detected from these seismic traces. We conclude that routine data-processing procedures, such as migration and multiple attenuation, have minor influence on singularity information. However, one should bear in mind that d) e) these comparisons are only relative because computing algorithms or noise could complicate simple comparisons. Li 共2002, 2004兲, in studies on the influence of random noise on singularity analysis, finds that the presence of noise generally lowers the average Hölder exponent. For a signalto-noise ratio larger than 10, the bias in the Hölder exponent does not exceed 15%, and the accumulated relative error is generally less than 0.5 共Li, 2004兲. Figure 7 illustrates the CWT result for the Kirchhoff-migrated seismic trace shown in Figure 6b and compares the results of estimating the Hölder exponent using WTMMP 共Figure 7b兲 and those using our linear method 共Figure 7d兲 at the Figure 7. 共a兲 Moduli of CWT coefficients of the synthetic seismic trace at CDP 600 in depth of 2155 m 共corresponding to point D in the Figure 6b. 共b兲 Moduli of CWT coefficients around depth 2155 m 共corresponding to point impedance trace in Figure 6c兲. Again, we found D in Figure 6c兲 and the modulus-maxima points. 共c兲 Estimation of the Hölder exponent from the linear least-squares regression of the modulus-maxima points in the double-logno big difference in the estimated Hölder expoarithmic space of scales versus squared moduli of CWT coefficients. 共d兲 Moduli of CWT nent between the two techniques. However, as coefficients around depth 2155 m. 共e兲 Estimation of Hölder exponent from the linear opposed to the linear double-logarithmic plots in least-squares regression using the points on the linear line directly above the singular Figure 2c and e, the squared moduli of CWT coefpoint 共shown in d兲. In both 共c兲 and 共e兲, Hölder exponents are estimated by using equation 18 from the slope of the linear least-squares regressions. ficients are curved on the logarithmic plots in Fig-
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ever, the picking on the scale range itself is a complicated issue because other factors, such as random noise 共often effective at smaller scales兲 and interference between closely spaced singular points 共mostly effective at larger scales兲 also can curve the train of data points in the double-logarithmic space of scales versus CWT coefficients. We therefore proceed with a simple approach that uses a uniform scale range corresponding roughly to the global frequency range of the seismic data for all singular points. Readers should bear in mind that for a low-resolution 共or local兲 singularity attribute as shown in Figures 1b, 4b, 5b, and 6a and e, sharp transitions in acoustic impedance may correspond with broad singularity lows in which the Hölder exponents change gradually. This circumstance might dim the true sharp transitions in acoustic impedance and/or might bias our interpretation. However, if readers are aware that low Hölder exponents correspond with sharp transitions in acoustic impedance, this kind of misinterpretation can be avoided easily. Moreover, the same problem can be largely avoided computationally by using a high-resolution singularity attribute 共Figure 8兲. Al-
though low-resolution 共or local兲 singularity attributes can be very useful in indicating local singularity behaviors of the amplitude envelopes of seismic reflections or acoustic impedances, high-resolution 共or pointwise兲 singularity attributes can faithfully capture the singularity patterns in the data at or around each data point. Highresolution 共or pointwise兲 singularity attributes can pinpoint exactly the singularity localities in data transitions, thereby avoiding the aforementioned problems. High-frequency singularity information residing in seismic data can be detected by using a mother wavelet of small scale. The scale is small enough so that in a uniform scale range of wavelet transform, the calculated Hölder exponents can characterize not only the singularities of sharp transitions but also those of more regular regions 共normally zero crossings in seismic data兲 between two adjacent singular points. One example of high-resolution singularity analysis of real seismic data is shown in Figure 8. The seismic section 共Figure 8a兲 is extracted from a 3D data set from the East China Sea Basin, which received primarily Cenozoic sediments of terrestrial facies. The Hölder section 共Figure 8b兲 is of high vertical resolution and shows the singularity information at or around each data point in the seismic data. High resolution is achieved because the Hölder exponent is a pointwise attribute, which not only can emphasize sharp transitions in the data but also can detect singular behaviors near zero crossings. Subtle stratigraphic transitions in seismic data, particularly near region A in Figure 8a, are enhanced by Hölder attributes. Minor faults also are illustrated better in Figure 8b. Furthermore, one can notice that the Hölder attribute is dependent not just on amplitude but also on the complicated interplay among seismic amplitude, phase, and frequency. For example, in Figure 8a, site B has higher amplitudes than site C, but on the Hölder section 共Figure 8b兲, site C shows higher singularity strengths by lower Hölder exponents because site C has higher frequencies. In this sense, the Hölder attribute can be regarded as a unifying attribute that simultaneously can convey important information about instantaneous amplitude, phase, and frequency with a single number and at each data point. Although the Marmousi2 model is a new-generation model with many improvements, it is inherently limited in the context of singularity because the geologic transitions are mostly step functions. It is hoped that later versions of the model will incorporate generalized transitions of various singularity strengths. But the model as it exists today has allowed us to further strengthen the connection between singularities in seismic data and impedance variations in the associated earth model. These are important results that support tests on real data reported elsewhere 共Li, 2002, 2004; Liner et al., 2004; Smythe et al., 2004; Li and Liner, 2005兲.
CONCLUSIONS Figure 8. 共a兲Aseismic line extracted from a 3D data set from the East China Sea Basin. Scale to right shows relative amplitude. 共b兲 Calculated high-resolution Hölder attribute. Scale to right shows the singularity exponent, which has no units. A, B, and C are sites for crossreferencing between seismic amplitude 共a兲 and Hölder attribute 共b兲. Hölder attributes give improved delineation on stratigraphic boundaries and at the same time characterize the complicated interplay among seismic amplitude, phase, and frequency, as well as singularity information in the subsurface acoustic impedances.
Both theoretical and numerical studies imply that surface seismic reflection data can carry rich singularity information related to acoustic impedance variations. A seismic trace can be taken as proportional to a smoothed version of the 共N Ⳮ 1兲th derivative of acoustic impedance, where N is the vanishing moment of the seismic wavelet. By applying wavelet-based multiscale singularity analysis, singularity strengths and localities in acoustic impedance can be detected directly from seismic reflection data. Experienced seismic interpreters can benefit from insights gained by this new attribute.
Wavelet-based detection of singularity Our numerical method is an approximation to the theoretical work 共equation 18兲, but it saves computing time and has proved to be effective in detecting both local 共low-resolution兲 and pointwise 共highresolution兲 singularities. Because singularity analysis is a relatively new arena in seismology, many aspects deserve further investigation. On the algorithm side, we note that potentially more accurate and expensive algorithms are known, such as 2-microlocal analysis and the slope-wavelet method. Applications of those algorithms of singularity analyses may improve the images of 2D sections or 3D volumes of the Hölder attribute. On the seismic modeling side, we would like to see more accurate modeling that incorporates generalized transitions in the elastic model, not just step functions as in the current Marmousi2 model. More generalized models and elastic synthetics might allow more direct inspection of the behaviors of singularity transmission from acoustic impedances to surface seismic data.
ACKNOWLEDGMENTS This research is funded by the Natural Science Foundation of China 共grant 40504016兲 and by the National Basic Research Program of China 共973 Program兲 共grant 2007CB411702兲. The Marmousi2 elastic model and synthetic data 共Martin, 2004兲, provided at the www. agl.uh.edu Web site, are particularly appreciated. We thank A. Gersztenkorn and two other anonymous reviewers for their very careful reviews and many important suggestions. We also extend our sincere appreciation to the associate editor, Dengliang Gao, whose critical and comprehensive comments helped significantly in improving the quality of this paper.
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