Interpretive advantages of 90 -phase wavelets: Part 1 Modeling

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1 GEOPHYSICS, VOL. 70, NO. 3 (MAY-JUNE 2005); P. C7 C15, 11 FIGS / Interpretive advantages of 90 -phase wavelets: Part 1 Modeling Hongliu Zeng 1 and Milo M. Backus 1 ABSTRACT We discuss, in a two-part article, the benefits of 90 - phase wavelets in stratigraphic and lithologic interpretation of seismically thin beds. In Part 1, seismic models of Ricker wavelets with selected phases are constructed to assess interpretability of composite waveforms in increasingly complex geologic settings. Although superior for single surface and thick-layer interpretation, zero-phase seismic data are not optimal for interpreting beds thinner than a wavelength because their antisymmetric thin-bed responses tie to the reflectivity series rather than to impedance logs. Nonsymmetrical wavelets (e.g., minimum-phase wavelets) are generally not recommended for interpretation because their asymmetric composite waveforms have large side lobes. Integrated zero-phase traces are also less desirable because they lose high-frequency components in the integration process. However, the application of 90 -phase data consistently improves seismic interpretability. The unique symmetry of 90 -phase thin-bed response eliminates the dual polarity of thin-bed responses, resulting in better imagery of thin-bed geometry, impedance profiles, lithology, and stratigraphy. Less amplitude distortion and less stratigraphy-independent, thin-bed interference lead to more accurate acoustic impedance estimation from amplitude data and a better tie of seismic traces to lithology-indicative wireline logs. Field data applications are presented in part 2 of this article. INTRODUCTION Zero-phase wavelets have long been considered the best interpretive wavelets for seismic interpretation. Wood (1982) outlines properties of zero-phase wavelets. Wood (1982) and Yilmaz (2001) address the principles of wavelet processing to replace a source wavelet with a zero-phase, interpretive wavelet without altering its amplitude spectrum. Brown (1991) summarizes many advantages of the zero-phase wavelet, including wavelet symmetry, minimal ambiguity of wavelet shape in correlation, center and maximum amplitude of the wavelet coinciding in time with reflection interface, and best resolution among wavelets with the same amplitude spectrum. The assumption under which zero-phase wavelets are deemed superior to other wavelets is that the reflection must come from a single interface, or seismic data must have enough resolution to resolve individual reflections from the top and bottom of a bed. Sengbush et al. (1961), Widess (1973), and Meckel and Nath (1977) analyze the issue of interface resolution of zero-phase wavelets and conclude that the resolution limit for zero-phase wavelets is about one-quarter of the dominant wavelength (λ/4). Below λ/4, the top and bottom of the bed can no longer be picked correctly in time, and reflections from the two surfaces interfere, resulting in composite waveforms. From λ/4 toλ, the top and bottom are better resolved, but composite waveforms still differ critically from the zero-phase wavelet. Most petroleum reservoirs are small in vertical dimension (<λ), and many are below seismic resolution (<λ/4). For interbedded thin reservoirs, we cannot resolve top and bottom interfaces for accurate bed geometry, and we regularly even have difficulties locating interfaces from interference-plagued composite waveforms. As a result, zero-phase wavelets, designed for optimal mapping of single interfaces, become obsolete. We are therefore more interested in detecting the entire thin bed as a composite event, which requires reconditioning seismic data with a different wavelet. Studies by Sicking (1982), Zeng et al. (1996, 2003), Zeng (2003), and Zeng and Hentz (2004) suggest that 90 -phase wavelets are a better choice for seismic thin-bed interpretation. A 90 -phase seismic volume is not only better for mapping reservoir geometry in an interbedded sandstone/shale section (Zeng et al., 1996; Zeng and Hentz, 2004), but it also better fits the geologists view of the subsurface by mimicking the impedance log (Sicking, 1982). These two companion papers discuss the interpretive advantages of 90 -phase wavelets. Part 1 covers the seismic Manuscript received by the Editor June 4, 2004; revised manuscript received September 8, 2004; published online May 20, The University of Texas at Austin, Bureau of Economic Geology, Austin, Texas hongliu.zeng@beg.utexas.edu; backus@mail.utexas.edu. c 2005 Society of Exploration Geophysicists. All rights reserved. C7

2 C8 Zeng and Backus modeling of simple and complex geologic models for basic principles and benefits of 90 -phase wavelets. Part 2 (Zeng and Backus, 2005) deals with field data applications that illustrate the value of 90 -phase data in lithology and porosity prediction, reservoir geometry mapping, stratigraphic correlation, and the study of depositional geomorphology. PHASE CONTROL OF WAVELET SHAPE In the frequency domain, a wavelet can be uniquely expressed by amplitude and phase spectra. The amplitude spectrum determines the energy distribution of different frequency components of the seismic signals; the phase spectrum displays phase lags of respective signals in the amplitude spectrum. Assuming the same amplitude spectrum, a linear-phase change leads to a time shift of a wavelet; a constant-phase shift alters wavelet shape. By modifying the slope of the linearphase shift and constant-phase shift, an infinite number of wavelets can be created from the same amplitude spectrum (e.g., Yilmaz, 2001). Figure 1 examines the constant phase shift of a Ricker wavelet obtained by applying the Hilbert transform (Bracewell, 1999). Starting with a symmetric, zero-phase Ricker wavelet, a90 -phase shift converts a zero-phase wavelet to an antisymmetric wavelet, whereas a 180 -phase shift reverses its polarity. A 270 -phase shift reverses polarity while making the wavelet antisymmetric again. Eventually, a 360 -phase shift moves the wavelet back to itself. As far as arrival time is concerned, the maximum amplitudes of zero-, 180 -, and phase wavelets are aligned at t = 0, whereas zero crossings of 90 - and 270 -phase wavelets are pointed at t = 0. Other intermediate changes of constant phase lead to nonsymmetrical wavelets (e.g., 30,60, 120, 150, 210, 240, 300,and 330 phase, Figure 1) that have no distinctive amplitude characteristics at t = 0. In particular, the 90 -phase wavelet is equivalent to the quadrature trace (Taner and Sheriff, 1977) if the original wavelet is zero phase. Other important seismic phase characteristics, such as minimum phase, mixed phase, and maximum phase, are caused by nonlinear phase distributions and are not discussed here in detail. A zero-phase wavelet can be converted to a minimumphase or maximum-phase wavelet having the same amplitude spectrum by applying spectral factorization (e.g., Claerbout, 1976). The interpretive quality of a wavelet is also related to the relative size of its side lobes. Smaller side lobes mean less ambiguity, fewer interferences, and higher accuracy in interpretation. In our synthetic experiments, we exclusively use the Ricker wavelet, which is characterized by a main- to side-lobe amplitude ratio of For field seismic data, the size of side lobes is controlled by seismic bandwidth (Yilmaz, 2001). The wider the bandwidth, the smaller the side lobes and the greater the ratio. Ignoring side lobes beyond the first side lobe, a octave (5/20/60/75) bandpass wavelet is equivalent to a 40-Hz Ricker wavelet in terms of the main- to side-lobe amplitude ratio, a condition easily satisfied for most modern seismic data. Therefore, observations from Ricker wavelet models should be applicable for seismic interpretation of reasonably wide bandwidth field data. WEDGE MODEL To assess the merits of 90 -phase wavelets in seismic thinbed interpretation, we first performed a simple model study of an isolated thinning bed. The purpose was to establish basic thin-bed waveform characteristics of selected wavelets and their interpretive implications. A two-bed interference model and a geologically more realistic interbedded thin-bed model are discussed in the following sections. Model design The fundamental model consists of a wedge of material encased in acoustically dissimilar material (Figure 2). The top and bottom of the wedge show opposite reflection polarities, a realistic and typical situation in most, if not all, stratigraphic profiles. We assume that sandstone wedge material is lower in acoustic impedance (AI) than the surrounding material (shale) to mimic the petrophysical profile of the shallow Gulf of Mexico and other areas. The reflection coefficient R, calculated from the impedance profile, is negative at the top and positive at the bottom with the same magnitude. The model covers a thickness range from seismically thin (<λ/4) to marginally thick (λ/4 λ) and seismically thick (>λ). Synthetics and amplitude tuning curves A zero-phase Ricker wavelet synthetic section of the wedge model is shown in Figure 3. The polarity convention of the zero-phase data adopted in this paper is that a negative reflection coefficient corresponding to a decrease in acoustic impedance is displayed as a trough. The composite waveforms of the wedge show dramatic changes with thickness. We observe symmetrical waveforms whose centers (maximum amplitudes) correspond to the top and bottom of the Figure 1. Wavelet phase control over wavelet shape and symmetry. Ricker wavelets of nonzero-phase lags are created by applying the Hilbert transform to the Ricker wavelet. Figure 2. Wedge AI model of sandstone encased in thick shale. Assuming constant AI in sandstone and shale, R is negative at the top and positive at the bottom, with the same magnitude; λ denotes seismic wavelength.

3 Advantages of 90 -phase Wavelets: Part 1 C9 sandstone when the sandstone is thicker than a wavelength (λ).however, forseismicallythin beds (<λ/4) and marginally thick beds (λ/4 λ), reflection amplitudes are composite seismic responses that mix reflections from the top and the bottom (Figure 3). In this situation, the observed waveform is such that the bed corresponds to an antisymmetric, trough-thenpeak couplet (Widess, 1973). When the bed is thinner than 3λ/4, the center of the bed is aligned at the seismic zero crossing. A90 -phase Ricker wavelet seismic model is displayed in Figure 4. For a 90 -phase wavelet, the polarity convention is chosen such that a negative-then-positive reflection coefficient set corresponding to a thin bed with acoustic impedance lower than in the host rock is displayed as a trough. This is equivalent to applying a 90 -phase shift to a zero-phase wavelet and then reversing the polarity. In this model, when the sandstone is thicker than λ, we see antisymmetric waveforms whose zero crossings align with the top and bottom of the sandstone. For seismically thin beds and marginally thick beds, reflection amplitudes are composite seismic responses (Figure 4, <λ). The observed waveform becomes symmetric so that the bed corresponds to a seismic trough event. When the bed is thinner than 3λ/4, the center of the bed is aligned at the center of the trough (maximum negative amplitude). If we measure composite amplitude for the zero-phase model and maximum negative amplitudes for the 90 -phase model, comparable thin-bed amplitude tuning curves can be seen (Figure 5). In both models, we can approximately calculate actual bed thickness by measuring peak-to-trough traveltime (zero-phase case, Figure 3) or traveltime between zero crossings of the main lobe (90 -phase case, Figure 4) if the bed is thicker than λ/4 or by linearly linking amplitude to the actual thickness if the bed is thinner than λ/4. As a result, the application of 90 -phase wavelets does not reduce interface resolution or detection power of seismic amplitude. Interpretive advantages of 90 -phase seismic data By comparing the zero-phase seismic model (Figure 3) and the 90 -phase model (Figure 4), we observe that 90 -phase data are more desirable for geologic interpretation of thin beds. Of course, these observations are restricted to noise-free data having simple wavelets of small, Ricker-like side lobes. First, the 90 -phase model has easier-to-interpret waveforms and simpler interpretive rules. In the zero-phase model (Figure 3), the seismic response to a thin bed is a troughthen-peak couplet. The top of the bed corresponds to a main seismic trough, whereas the bottom of the bed can be correlated to a main seismic peak. When the bed thins to below λ/4, a time deviation between reflection interfaces and maximum seismic energy (trough or peak) occurs. To use seismic amplitude for geometric and lithologic interpretation, we may choose to deal with dual polarities (trough and peak) at the same time. However, doing so hampers an interpreter s ability to identify the trough/peak couplet associated with a given thin bed, potentially causing confusion. To use trough or peak amplitude alone may create a greater error when thin-bed interference of other geologic units is involved (as discussed in the following section). On the other hand, in the 90 -phase model (Figure 4), the seismic response to a thin bed is a single main trough. The center of the bed lines up with the maximum Figure 3. Zero-phase Ricker seismic model. Despite the symmetric shape of the Ricker wavelet, composite waveforms are symmetric only in seismically thick beds (>λ). Antisymmetric seismic responses dominate when a bed thins. If the bed is thinner than λ/4, deviation between reflection interfaces and seismic trough peak measurements (indicated by dashed lines) occurs. Neither polarity nor amplitudes in asymmetric waveforms match wedge geometry (lithology). Figure 4. Ninety-degree Ricker seismic model. Regardless of antisymmetric wavelet, composite waveforms become symmetric when thickness is less than λ. Although not very accurate when thickness is less than λ/4, the seismic trough (negative amplitudes) matches wedge geometry, with zero crossings corresponding to the boundaries of the wedge. Figure 5. Comparison of amplitude tuning curves for zero- and 90 -phase seismic models. Two models exhibit similar tuning curves with the same tuning thickness and similar apparent thicknesses measured from traveltime.

4 C10 Zeng and Backus negative amplitude (<3λ/4) or with the center of the composite trough waveform (3λ/4 λ). Although not very accurate if the thickness is less than λ/4, the top and bottom of the bed correspond to seismic zero crossings. If side lobes are ignored, seismic polarity (negative amplitudes, in this case) becomes a unique indicator of the thin bed. A simple highlight of (negative) amplitudes is adequate for depicting geometry of the thin bed. A 90 -phase shift converts a surface (reflectivity)- responsive signal (zero phase) into a more layer-responsive signal (90 phase). This conversion greatly reduces ambiguity in associating seismic events to thin beds, simplifying seismic interpretation rules. Second, the 90 -phase model is a better representation of impedance profile and lithology-indicative wireline logs. When the bed is thinner than λ, we hope to tie the seismic waveform directly to the impedance profile for the best possible correlation between seismic traces and stratigraphic architecture. If impedance is linked to lithology, as in our geologic model (Figure 2), we also hope to tie seismic waveforms to lithology-indicative wireline logs (e.g., gamma-ray and spontaneous potential logs in a sandstone/shale sequence). In a zero-phase model, however, the same lithology of the same impedance can be tied to opposite polarities. The correlation between impedance curves in the model and seismic traces is typically very poor. The 90 -phase model corrects the problem (Figure 4). Seismic responses are symmetric to the sandstone bed of symmetric impedance curves, making the main seismic event (a trough, in this case) coincide with the geologically defined sandstone bed. As a result, seismic polarity (negative amplitudes, in this case) is tied uniquely to lithology, and the correlation between impedance curves and seismic traces is greatly increased. This improvement should make a seismic section look more like a geologic section, encouraging more geologists to be interested in seismic interpretation. Minimum phase and integrated zero phase Minimum phase has been widely discussed in the literature. We apply the minimum-phase equivalent of a Ricker wavelet to generate a seismic wedge model (Figure 6). Comparing zero-phase and 90 -phase models (Figures 3 and 4), we find the biggest difference in seismic waveform is the loss of symmetry. With a nonsymmetrical wavelet, the am- plitude difference in main lobes and side lobes is reduced. Large side lobes make it more difficult to identify geologic surfaces (in this case, top and bottom of sandstone in Figure 6). Also, amplitudes from surfaces with the same magnitude of reflectivity are typically not identical. As a result, it is considerably more challenging to link amplitudes to thickness and lithology. Maximum- and mixed-phase data have similar problems. Although minimum-phase and maximum-phase wavelets may have advantages if geologic interference is derived mainly from one direction (below and above the layer, respectively) (Zeng et al., 1996), nonsymmetrical wavelets are less desirable interpretive wavelets and generally should be avoided. Another common approach is to integrate zero-phase seismic traces for an estimate of thin-bed AI. A zero-phase seismic trace is an estimate of reflectivity series; reflectivity can be approximated by the derivative of AI. Integrating the estimated reflectivity series (Figure 3) results in an estimated AI profile (Figure 7). However, the integration process reduces dominant and high-end frequencies of seismic traces, thereby reducing seismic resolution. Integrated traces are characterized by longer main lobes (at 0 λ/2) and side lobes (at any thickness) compared with the 90 -phase data (Figure 4) that preserve the same frequency components of zero-phase seismic data. As a result, integrated zero-phase data are less desirable for seismic interpretation. TWO-BED INTERFERENCE MODEL We further examine thin-bed interference patterns by modeling two low-ai thin sandstone beds inserted in a high-ai shale host (Figure 8). To simplify the discussion, thicknesses of the thin beds are assumed equal and fixed (λ/4, Figure 8a,b). Zero-phase (Figure 8a) and 90 -phase (Figure 8b) seismic models demonstrate more complex reflection patterns than do their single-wedge counterparts (Figures 3 and 4, respectively). Interference patterns For interpretive purposes, the simplest way to characterize interference patterns is to study seismic amplitude versus distance between thin beds. For the zero-phase model (Figure 8a), peak (0 P), trough (0 T), and trough-to-peak composite amplitude (0 TP) can be measured for upper and Figure 6. Minimum-phase seismic model. The minimum-phase wavelet is converted from a Ricker wavelet of the same amplitude spectrum as in Figures 3 and 4 by applying a spectral factorization. The minimum-phase model is characterized by asymmetric waveforms with large side lobes that cause ambiguity in interpreting geologic surfaces and lithology. Figure 7. Integrated version of zero-phase seismic model (Figure 3). Although the model visually resembles the 90 - phase seismic model (Figure 4), it is characterized by a fatter main trough and larger side lobes (peak).

5 Advantages of 90 -phase Wavelets: Part 1 C11 lower sandstones, respectively (Figure 8c,d). For the 90 - phase model (Figure 8b), trough amplitude (90 T) is a suitable measurement (Figure 8c,d). For comparison, all amplitudes in the zero-phase and 90 -phase models are normalized to respective single-bed responses. Four interference patterns are observed as a function of the shale wedge thickness: Amplitude distortion and impedance estimation Seismic interference causes amplitude distortions in both models, although with different magnitudes and potential interpretation pitfalls. In the zero-phase model (Figures 8a and 1) Interference free. If the shale is thicker than 3λ/4, there is little amplitude change compared with single-bed responses. If the shale bed is thicker than λ, amplitudes have no changes and single-bed tuning curves (Figure 5) apply. 2) Constructive interference. If the shale ranges roughly from λ/4to3λ/4, amplitudes are above the normal level. 3) Destructive interference. If the shale ranges from λ/8 to λ/4, amplitudes are below the normal level. 4) Nontraceable. If the shale is thinner than λ/8, composite waveforms for the two sandstones merge. No separate trough-to-peak couplets (zero phase) or troughs (90 phase) can be recognized for individual sandstones in the vertical section, although horizontal imaging by slicing a 3D volume is still possible. To further demonstrate the influence of interference patterns on stratigraphic and impedance interpretation, some variably spaced sandstone patches are added to a single sandstone model to simulate representative constructive, destructive, and nontraceable interference patterns identified in Figure 8 (Figure 9). Bed tracing and stratigraphic interpretation In the zero-phase model (Figure 8a), both upper and lower sandstone beds can be traced as trough-then-peak couplets if the beds are adequately separated (shale wedge >λ/4) despite constructive interference. If sandstone beds are close (shale wedge ranging from λ/8 toλ/4), however, uneven destructive interferences of troughs and peaks severely damage the waveform symmetry, and tracing of the reflection couplets may become confusing (Figures 8a and 9a). If sandstone beds are too close to each other (shale <λ/8), two couplets are reduced to one, and bed tracing of individual sandstones in a vertical section becomes impossible. If we attempt to trace continuous sandstone in the model in destructive and nontraceable interference ranges by slicing against a seismic reference (here, the generally continuous peak or trough), polarity reversals may occur (Figure 9a). These false polarity reversals are particularly harmful for stratigraphic analysis by creating pseudoseismic events that would be wrongly interpreted as additional depositional units or stratigraphic relationships. As a result, we potentially see more depositional bodies and erosional surfaces than actually exist or are resolved. In the 90 -phase model (Figures 8b and 9b), waveform symmetry as observed in the single wedge model (Figure 4) is not lost. As a result, delineation of the sandstones is much simpler. A trace of troughs recovers the accurate geometry of sandstone beds. Polarity reversals are rare and occur only in the nontraceable interference range, ensuring qualitatively correct horizontal amplitude patterns for closely spaced sandstone beds. This accuracy offers a clear advantage in the study of seismic stratigraphy and geomorphology, as shown in examples in part 2 of this article (Zeng and Backus, 2005). Figure 8. Interference patterns of two approaching thin beds encasing shales. AI is assigned the same as it is in Figure 2. Beds are equally thick at λ/4. (a) Zero-phase synthetic section. (b) 90 -phase synthetic section. (c) Amplitude versus thickness of the shale wedge for the upper sandstone. (d) Amplitude versus thickness of the shale wedge for the lower sandstone: 0 T = trough (negative) amplitude in the zero-phase data, 0 P = peak (positive) amplitude in the zero-phase data, 0 TP = trough-to-peak composite amplitude in the zero-phase data, and 90 T = trough amplitude in the 90 -phase data. Four interference patterns are identified as interference free (>3λ/4), constructive interference (λ/4 3λ/4), destructive interference (λ/8 λ/4), and nontraceable (<λ/8). If amplitudes are traced from troughs and peaks separately, constructive and destructive interferences for zero-phase data are different for the two beds and are therefore location (stratigraphic position) dependent.

6 C12 Zeng and Backus 9a), measured amplitude is a function of stratigraphic position. For the upper sandstone in Figure 8c, the seismic trough, which corresponds roughly to the top of the sandstone, undergoes minimal amplitude distortion. With shale wedge thinning, the trough amplitude (Figure 8c, 0 T) decreases slightly ( 1.34 db, λ λ/8). However, the seismic peak, which ties roughly to the base of the upper sandstone, suffers much more severe distortion (Figure 8c, 0 P). Amplitude first increases significantly (1.15 db in λ λ/4), then decreases radically ( db in λ/4 λ/8). For the lower sandstone (Figure 8d), the interference pattern is just the opposite: the trough amplitude (Figure 8d, 0 T) is distorted much more than the peak amplitude (Figure 8d, 0 P). Figure 9c illustrates similar relationships. This dependency of amplitude on stratigraphic position is obviously not desirable for interpretation, especially if we want to link amplitude to impedance contrast, lithology, and stratigraphy. To reduce amplitude distortion and eliminate amplitude dependency on stratigraphy, we can replace the measurement of trough or peak amplitude with composite amplitude (Figure 8c,d, 0 TP, from 0.52 to 4.14 db). However, this approach would be difficult to apply in field data, in which trough peak couplets may be tricky to pick in closely spaced thin beds without dense well control. The 90 -phase data are less plagued by amplitude distortion problems. The interference pattern is the same for both upper and lower beds (Figure 8b), so there is no stratigraphic dependency of amplitude. Amplitude distortion is mild (0.22 db in λ λ/2and 3.47 db in λ/2 λ/8) and remains the same for both sandstone beds (Figures 8c, 90 T, and 8d, 90 T). As a result, 90 -phase data are generally better correlated to acoustic impedance. Note that in this model, thin beds are fixed in thickness. In fact, the seismic interference pattern should change with bed thickness. However, general relationships observed in this model should hold if two beds are roughly equally thick and total thickness does not exceed 3λ/4. Conclusions should also apply to cases of more than two thin beds, with minor modifications from additional modeling. REALISTIC MODEL OF INTERBEDDED SANDSTONES AND SHALES Now we expand seismic modeling to a geologically realistic case. Mapped from well data, multiple interbedded sandstones and shales are modeled with geologically reasonable thickness and impedance distributions. Isolated beds and closely spaced beds are included, with variable magnitude of reflectivity at each depositional surface. Impedance is assumed to be an indicator of lithology (low AI as sandstone and high AI as shale), although more ambiguous situations may occur in field data applications. Geologic and impedance models Zeng et al. (1996) have built a fine-scale Miocene geologic model based on depositional facies analysis and facies-guided property mapping in the Powderhorn field, Calhoun County, Texas. More than 100 wells were used to constrain thickness and clay-content (C clay ) mapping of 14 subsurface sandy depositional units (labeled 1 14, Figure 10a). A stack of these sandy depositional units and interbedded shale units compose a 3D model. The model is geologically realistic in the sense that all rock properties mapped (thickness, C clay, porosity, AI, etc.) honor the facies interpretation of fluvial and microtidal shore-zone systems in the formation. In this 200-ms (300 m, assuming an average velocity of 3000 m/s) section, most of the beds are less than 15 ms (<22 m) thick, well within the range of seismic interference with a typical seismic wavelet in the area (3λ/4 = 28 m at 40 Hz dominant frequency). Vertical variation of rock properties within beds (or in transitional zones) was not modeled. Figure 9. Typical amplitude distortion of a thin bed caused by one other thin bed nearby at different interference patterns as defined in Figure 8. (a) Zero-phase synthetic section. (b) 90 -phase synthetic section. (c) Amplitude distortion at nontraceable, destructive, and constructive interference patterns. Amplitudes are measured at surfaces made by linearly interpolating between maximum troughs or peaks where only one sandstone exists. Potential for polarity reversal in a horizontal amplitude display is seen easily in the zero-phase data. Seismic models and interpretation Assuming low-impedance sandstones in high-impedance shales and a linear relationship between sandstone AI and shale AI with C clay (Han et al., 1986), AI distribution is a scaled version of the C clay model (Figure 10a). For simplicity, the low-frequency, vertical trend of AI is not modeled. By convolving 40-Hz, zero-phase, and 90 -phase Ricker wavelets

7 Advantages of 90 -phase Wavelets: Part 1 with the normal-incidence reflectivity model calculated from the acoustic impedance model, we obtain respective seismic models (Figure 10b,c). For seismically thin (<λ/4), sandy stratigraphic units encased in thick shale (e.g., 1 3, 9, and 12 in Figure 10b,c), improvement can be seen in impedance and lithology profiling but not in geometry (thickness) interpretation with 90 -phase data. Amplitudes in 90 -phase data (Figure 10c) clearly tie to geologically defined impedance layers or lithology (trough C13 for lower AI/sandstone and peak for higher AI/shale), which is not the case when zero-phase data (Figure 10b) are used. However, trough-to-peak couplets of these thin beds in zerophase data can be traced reliably as long as the stratigraphic architecture is known. Picking zero crossings in the 90 -phase data produces similar thickness estimation. For seismically thin (<λ/4), sandy stratigraphic units interbedded with thin shales (e.g., 6 7 in A, 5 7 in B, and 9 10 in C, Figure 10), 90 -phase data are a better choice. Similar to observations in two-bed interference models (Figures 8 and 9), closely spaced thin beds are difficult to trace in zero-phase data because of the loss of amplitude symmetry of trough peak couplets. This difficulty can be avoided, however, by following seismic troughs in 90 -phase data. Amplitude distortion is also greater and is stratigraphy dependent in the zerophase data, typically leading to a poorer estimate of acoustic impedance. For marginally thick sandstones (λ/4 λ), 90 -phase data still show a better fit to the lithologic profile, thanks to the symmetry of the seismic response (e.g., D in 14; compare Figure 10b and 10c). For seismically thick (collective thickness >λ) units (e.g., E in 5 7, F in 13 14, Figure 10b,c), both data sets show separate reflections corresponding to the top and bottom of the beds. Although traveltime can be measured for unit geometry, amplitude and polarity are not a direct indicator of bed geometry. However, zero-phase data are better because the waveform is symmetric to the bed boundaries. Crosscorrelation between AI profile and seismic amplitude Quantitative comparison between zero-phase and 90 phase data can be achieved by crosscorrelating between the AI model and respective seismic models (Figure 11). For 40Hz synthetics, 90 -phase amplitude traces show the best correlation to the AI model without time shift (correlation coefficient ρ = 0.73, Figure 11b). Zero-phase amplitude traces are poorly tied to the AI model at zero time shift (ρ = 0.01, Figure 10. Clay content (Cclay ), AI, and seismic models in a geologically realistic setting; (a) Cclay and AI section; Cclay and AI are linearly related, with low AI indicating low Cclay (sandstone) and high AI denoting high Cclay (shale). (b) 40Hz, zero-phase Ricker synthetics. (c) 40-Hz, 90 -phase Ricker synthetics. A, B, and C refer to closely spaced thin beds (individual beds thinner than λ/4) with destructive interference pattern (Figures 8 and 9). D denotes a marginally thick bed (λ/4 λ). E and F are seismically thick beds (total thickness >λ). Numbers 1 4, 8, 10 11, and 13 are fluvial facies; 5, 9, and 12 are wave-dominated delta facies; and 6, 7, and 14 are barrier/bar-lagoon facies. Dashed lines are geologically defined unit boundaries and impedance surfaces. Figure 11. Crosscorrelation functions between the AI model (Figure 10a) and seismic models of selected frequencies and phases. Compared with zero-phase cases, 90 -phase data with dominant frequencies of 20, 40, and 80 Hz are all better correlated to the AI model, even if the zero-phase data are properly shifted in time. The AI logs (A, B, C) and thickness AI logs (D, E, F) are treated separately to show the influence of thickness tuning.

8 C14 Zeng and Backus Figure 11b), but the correlation improves dramatically (ρ = 0.66) when the zero-phase model is shifted λ/4 downward so that the seismic troughs are best tied to sandstones in AI logs. However, overall correlation is still poorer than that in 90 - phase data results. Crosscorrelation functions from 20- and 80-Hz models exhibit similar trends. The quality of correlation, however, deteriorates for both zero- and 90 -phase data. The 40-Hz model has the best correlation because most thin beds in the model are in the range of constructive interference for the 40-Hz data but are in the range of destructive interference for the 20-Hz data and are the seismically thick beds for the 80-Hz data (refer to Figure 8c,d). Increased amplitude distortion in the 20-Hz data and the greater number of resolved surfaces in the 80-Hz data reduce AI correlation to the amplitude. A more precise treatment considers thickness tuning. Reducing data frequency places more beds into a seismically thin (<λ/4) category, with enhanced (linear) amplitude-thickness correlation (Figure 5). As a result, replacing AI logs with thickness-weighted AI (thickness AI) logs improves crosscorrelation in 20- and 40-Hz data (Figure 11d,e) but reduces correlation in the 80-Hz case (Figure 11f). DISCUSSION Seismic modeling suggests that different wavelet phases can be designed for different geologic objectives. Neither the zero phase nor the 90 phase is universally better, and neither is better in some important cases. Whenever we are dealing with a well-resolved interface reflection, all published advantages of the zero-phase wavelets hold true. Also, if we are interpreting a major unconformity or a crisp fault interface reflection, zero-phase wavelets have advantages, although a good case can be made for use of a minimum-phase or maximum-phase wavelet under certain conditions (e.g., Zeng et al., 1996). In all cases dealing with seismically thin beds, 90 -phase wavelets have advantages. In this work we have examined low-ai sandstones (relative to shale) only. Low-AI sandstones are typical among shallow formations with minor compaction, especially marine sediments in the Gulf of Mexico and elsewhere. However, other types of sandstone shale acoustic relationships also have significant impact on the seismic behavior of any given wavelet. In hard-rock onshore sediments or deeply buried marine sediments, high-ai sandstones (relative to shale) are more representative. Compared with the low-ai sandstone relationship, there is a reversal in AI contrast and resultant seismic polarity. Therefore, observations made from our low-ai sandstone models should hold true if reversed polarities are applied to seismic models. In seismic interpretation of thin reservoirs, another relevant issue is how to characterize a bed bounded above and below by acoustically dissimilar rocks or an acoustically transitional bed. All discussions so far have assumed equal-magnitude reflectivity at the top and bottom of a thin bed, implicating blocky AI layering without vertical AI variation in the host rocks and within the unit. Many times, vertical changes of AI in the host rocks or within the thin bed occur because of depositional architecture and resultant grain-size trend variations. We have observed significant waveform and phase changes caused by AI variations in host rocks and within thin beds in model and field seismic data. Further investigation is needed to evaluate how the seismic-phase property influences characterization of such thin beds. CONCLUSIONS Despite its superiority in single-interface and thick-bed interpretation, the zero-phase wavelet is less suitable than the 90 -phase wavelets for interpretation of seismically thin (<λ/4) and marginally thick (λ/4 λ) beds. Disadvantages include a poor match of seismic traces to AI logs and detrimental seismic interference in a thinly interbedded environment. Minimum-phase and other nonsymmetrical wavelets are also not recommended because of ambiguity caused by large, asymmetric side lobes. Seismic modeling of a thinning bed reveals that a 90 -phase seismic trace tends to be a good estimate of relative AI log. The application of 90 -phase wavelets leads to a better tie between seismic events and impedance/lithology layering while maintaining an amplitude-tuning relationship similar to that of zero-phase data. Integration of seismic traces can convert zero-phase data to fit the impedance profile, but the process enhances low-frequency energy and reduces seismic resolution. Seismic interference patterns in a model of converging thin beds further illustrate that a 90 -phase shift reduces polarity reversals and amplitude distortion in thin-bed imagery compared with zero-phase data. The 90 -phase data also eliminate dependency of amplitude on stratigraphy. As a result, accuracy and resolution of impedance, lithology, and stratigraphic correlation can be improved. Finally, 90 -phase seismic models of an interbedded sandstone/shale model demonstrate consistent improvement in stratigraphic correlation and lithologic mapping over zerophase data in a geologically realistic setting. Crosscorrelation between the AI model and the seismic amplitude section is higher for 90 -phase data than for zero-phase data for a wide range of data frequencies. ACKNOWLEDGMENTS The authors thank S. Fomel and X. Janson at the Bureau of Economic Geology, the Geophysics associate editor, reviewer W. T. Wood, and two anonymous reviewers for their helpful comments and technical input. The manuscript was edited by L. Dieterich before submission. Graphics were drafted with the assistance of J. Ames. Partial support for this publication was received from the John A. and Katherine G. Jackson School of Geosciences and the Geology Foundation of The University of Texas at Austin. Published by permission of the Director, Bureau of Economic Geology. REFERENCES Bracewell, R. N., 1999, The Fourier transform and its applications, 3rd ed.: McGraw-Hill Book Company. Brown, A. R., 1991, Interpretation of three-dimensional seismic data, 3rd ed.: AAPG Memoir 42. Claerbout, J. F., 1976, Fundamentals of geophysical data processing: Blackwell Scientific Publications, Inc. Han, D. H., A. Nur, and D. Morgan, 1986, Effects of porosity and clay content on wave velocities in sandstones: Geophysics, 51,

9 Advantages of 90 -phase Wavelets: Part 1 C15 Meckel Jr., L. D., and A. K. Nath, 1977, Geologic considerations for stratigraphic modeling and interpretation, in C. E. Payton, ed., Seismic stratigraphy: AAPG Memoir 26, Sengbush, R. L., P. L. Lawrence, and F. J. McDonald, 1961, Interpretation of synthetic seismograms: Geophysics, 26, Sicking, C. J., 1982, Windowing and estimation variance in deconvolution: Geophysics, 47, Taner, M. T., and R. E. Sheriff, 1977, Application of amplitude, frequency, and other attributes to stratigraphic and hydrocarbon determination, in C. E. Payton, ed., Seismic stratigraphy: AAPG Memoir 26, Widess, M. B., 1973, How thin is a thin bed? Geophysics, 38, Wood, L. C., 1982, Imaging the subsurface, in K. C. Jain and R. J. P. defigueiredo, eds., Concepts and techniques in oil and gas exploration: SEG, Yilmaz, O., 2001, Seismic data analysis, 2nd ed.: SEG. Zeng, Hongliu, 2003, Significance of seismic phase in interpretation of stratigraphy and sedimentology: Convention, Canadian Society of Petroleum Geologists/Canadian Society of Exploration Geophysicists, Abstract, 266S0131. Zeng, Hongliu, and Milo Backus, 2005, Interpretive advantages of 90 -phase wavelets: Part 2 Seismic applications: Geophysics, 70, C17 C24. Zeng, Hongliu, and T. F. Hentz, 2004, High-frequency sequence stratigraphy from seismic sedimentology: Applied to Miocene, Vermilion/Block 50 Tiger Shoal area, offshore Louisiana: AAPG Bulletin, 88, no. 2, Zeng, Hongliu, M. M. Backus, K. T. Barrow, and N. Tyler, 1996, Facies mapping from three-dimensional seismic data: Potential and guidelines from a Tertiary sandstone shale sequence model, Powderhorn field, Calhoun County, Texas: AAPG Bulletin, 80, no. 1, Zeng, Hongliu, S. C. Ruppel, and R. Jones, 2003, Reconditioning seismic data for improved reservoir characterization, lower Clear Fork and Wichita, Fullerton field, West Texas, in T. J. Hunt and P. H. Lufholm, eds., The Permian Basin: Back to basics: West Texas Geological Society Publication ,

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