In this work, we show an example of the joint application of ings. These attributes Since a long time used to supply us detect geometry of the expected channel in the study. Still, the FFT by important information and details about the geologic features, li- technique of spectral decomposition has diligently detected the pos- thology, and stratigraphic channels Taner, ; Chopra and Marfurt, sible, channels geometry, and associated reservoir lithology. However, the most important im- plementation of seismic attributes is to acquire information from the 1.
Study area raw seismic data which is not easily obvious Anees, However, the evaluating 65 km. Therefore, and Perhaps it is a favorite way company Fig. Please cite this article as: Othman, A. Othman et al. Materials and methods The study has 3D post-stack seismic volume acquired in and Sedimentation of the Pliocene in the Nile Delta consists of a deep processed by PGS in over a total area of km2 Fig. These marine se- conventionally interpreted using the integrated Petrel reservoir char- quences are the shaly sediments of the Kafr El-Sheikh Formation.
The strati- Upper channel sand, lower channel Sand. Where the lower channel appears as low morphology with this attribute Figs. This indicator on the east part may be connected with the upper channel. Notice the average energy for the lower channel is relatively high geomorphology Fig.
Spectral decomposition Castagna et al. Compiled stratigraphic column of the study area. Petrobel internal re- beneath hydrocarbon reservoirs. Peyton used coherency and port Partyka et al. Upper Channel is younger and incises into lower Channel. Hardy et al. The changed outcomes in- failed to fully describe the channel extension. The Seismic cubes. Results and discussion Prior to running the spectral decomposition, the seismic data peak frequency is about 25 Hz, and the frequency band is from 10 Hz to According to the previous seismic attribute maps along the two 80 Hz Fig.
Amplitude Spectrum for the full seismic data within the reservoir interval. Average energy map at depth m upper channel in Temsah area. In this way we outline, that reservoir dis- color, and 65 Hz the highest frequency range in the seismic dataset in semination is within low frequencies and high amplitudes. On the Blue color Figs. In this way integrating the AAA map and energy attribute.
The dim parts at the middle part of tuning cubes with common frequency cubes is fundamental for exact the Upper channel on the AAA map Fig. EcoMan has negotiated special rates for a limited number of rooms in the hotel. Early registration will help to secure a room at the reduced rate. ECOMAN reserves the right to alter the content, location of the seminar or the identity of the speakers in case of events beyond our control. This special discount will apply to the 10th paid nomination onwards on programs offered during this year. In addition to the above plan, one extra free place is offered to any organization that makes 2 paid nominations for the same program and dates.
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The energy values corresponding to the salt are higher than the surrounding areas, and the salt body can be detected more effectively from the energy volume than by using a seismic volume. Use of a seismic volume would result in bleeding across the boundaries and would prevent a crisper definition of the salt canopy. After Gao Attempts to analyze seismic facies usually involve two steps.
First, seismic facies patterns are defined in terms of their lateral and vertical extents.
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Second, those defined seismic facies are interpreted in terms of their lateral and vertical associations and their calibration with wells, all of which give insight into the geological and depositional settings. This step is Significant because the relationship between seismic data, seismic facies, and depositional environment is not unique. Conventionally, an interpreter delineates seismic facies between mapped horizons.
This entails examining the dominant seismic facies on vertical sections through the seismic volume and posting that information on a map. Usually, a 2D map is produced that generalizes the distribution of seismic facies vertically within a mapped interval. Besides being laborious, it may be difficult to map different seismic facies consistently in large and complex areas, especially if multiple mapping units are involved. West et al. Tn West et al. The interpreter selects polygons on key seismic lines extracted out of 3D seismic volumes that exhibit different seismic facies.
The computer then computes the associated GLCMs, and in this way several examples of each class of facies are specified. After the training procedure, several quality controls are run. The result of this exercise is a seismic facies classification volume in which each trace and sample has a seismic facies classification. Figure 3 J shows the definition of training polygons on a seismic section. The interpreter has selected the polygons on the basis of their reflection character.
The result of the textural analysis is a seismic classificauon volume, which is examined and calibrated with the available well and core information. Figure 32b is an example from a. Because the textural-analysis seismic-facies classification is a volume, this type of analysis can be applied at different stratigraphic levels within an interval of interest, whether or not the interval is bounded by mapped horizons. After West et al. Chapter Summary Because of the oscillatory nature of the seismic wavelet, and hence of the seismic trace, all good measures of changes in reflector amplitude should be calculated along the dip and azimuth of an assumed reflector.
Lateral changes in bed thickness, lithology, and porosity result in lateral changes in thin-bed tuning. Eigenstructure and crosscorrelation coherence algorithms are designed to be sensitive only to reflector waveforms and will not see coherent lateral changes in reflector amplitude. Other coherence algorithms, including semblance, variance, and Manhattan-distance estimates of reflector similarity, are sensitive to both amplitude and waveform.
In a similar manner, algorithms designed to measure changes in reflector amplitude - such as Luo et a1. Unfortunately, all such algorithms commonly are referred to as coherence or edge detectors, which further obscures their differences. Whenever possible, we recommend using algorithms that are as mathematically decouplcd from each other as possible, thereby providing the interpreter with orthogonal independent views of his or her data.
Obvious end members exist that we could separate. For instance, a lateral change in porosity of a thin reservoir may give rise only to a subtle amplitude variation, with the waveform remaining constant. In contrast, a continuous reflector may be crosscut by backscattered ground roll. The amplitude of the coherent part of the reflector would remain constant, whereas the waveform and coherence measure would not. In contrast, energy-weighted coherent-amplitude gradients are sensitive to lateral changes in amplitude for a fixed waveform.
Such gradients are particularly effective in detecting thin channels that are not seen by eigenstructure and principal-component coherence algorithms, and the gradients provide insight into reservoir heterogeneity. Because they work only on the coherent component of the data, energy-weighted coherent-amplitude gradients highlight channels that have stronger reflectivity and deemphasize incoherent fault zones that may confuse our stratigraphic interpretation.
Long-wavelength estimates of amplitude variability, such as Luo et aJ. Longwavelength second-derivative estimates of amplitude variability, achieved by applying the most-positive-curvature and most-negative-curvature algorithms to amplitude gradients, also can enhance such subtle changes in reflectivity. At present, long-wavelength estimates of changes in amplitude should be applied with care. Reflector dip varies slowly in the vertical direction except near unconformities , thereby allowing long-wavelength estimates of structural curvature on time slices.
In contrast, seismic amplitude varies rapidly in the vertical direction and renders long-wavelength estimates of amplitude changes challenging in structurally complex terrains. Although long-wavelength derivative calculations can be applied readily to dip-. The coherence slice Figure 32a highlights the lateral edges of the channel red lines , a broad, older sinuous element I , and a narrower, younger sinuous element 2. An interesting fact is that because the textural analysis seismic facies classification yields a volume, this type of analysis can be applied at different stratigraphic levels in an interval of interest, regardless of whether those intervals are bounded by mapped horizons.
Calibration of different features with the well data results in an interpretation map of the environment of deposition Figure 32c. The high-energy feature red along a channel system is revealed clearly by rendering the low-energy amplitudes as transparent, The same would be difficult to visualize and isolate from the original amplitude volume.
A horizon slice comparison at the same stratigraphic level from a a texture energy volume and b a seismic amplitude volume. Notice how the channel and levee deposits can be recognized, mapped, and detected more effectively from the texture energy volume than from the seismic amplitude volume. A typical seismic facies classification using an interpreter-trained probabilistic neural network, in which multiple facies classes have been identified.
The seismic classification scheme on the right consists of high amplitude HA , moderate amplitude MA , low amplitude LA , continuous C , and semicontinuous SC seismic facies. For that reason, we recommend that long-wavelength estimates of amplitude gradients be restricted to subvolumes of data that have been flattened to remove most of the structural curvature. Texture-attribute studies usually are not associated with seismic attribute studies. We consider texture analysis in its most general form to be a superset of the geometric attributes that comprise the major focus this book. Texture analysis uses second-order statistics to extract attributes that in turn help describe the textural properties of classes.
Textures based on GLCMs work well so long as the granularity of textures being examined is of the order of the pixel size, which commonly occurs on seismic data. On the basis of our analysis, we conclude that I. References Adeogba, A. McHargue, S. Blumentritt, C. Sullivan, and K. Chopra, S. Gao, D. Haralick, R. Shanmugam, and I. Luo, Y, S.
Marhoon, and M. Alfaraj, , Generalized Hilbert transform and its application in geophysics: The Leading Edge, 22, Luo, Y. Higgs, and W. Kirlin, , 3-D broadband estimates of reflector dip and amplitude: Geophysics, 65, Mitchum, R. Partyka, G. Reed, T. Reilly, J. West, B. May, J.
Eastwood, and C. Rossen, , Interactive seismic facies classification using textural and neural networks: The Leading Edge, 21, Widess, M. Chapter Objectives After reading this chapter, you will be able to identify the geologic features highlighted by spectral decomposition. Introduction Since the beginning of digital recording, geophysical data processors have decomposed the measured seismic signal into its Fourier or spectral components to attenuate low-frequency ground roll, or Hz cultural noise, and high-frequency random noise.
Also, data processors have routinely balanced the source spectrum through seismicdeconvolution and wavelet-shaping techniques to account for the input source signature, spectral changes resulting from ghost-period multiples, and attenuation of the overburden. Any time series can be represented in terms of a summation of other time series. For example, in Fourier analysis, we can represent any time series by a weighted summation of selected sinusoidal functions. In this example, the set of the selected sinusoidal time series is termed basis functions because they are the units from which we can recreate the original time series.
If one of those sinusoidal functions is crosscorrelated with another one of a different selected frequency, the crosscorrelation will be zero. Mathematicians would state that such selected sinusoidal functions are orthogonal perpendicular to each other. Thus, such a set of sinusoidal functions is not internally redundant.
In recreating the original time series, no particular sinusoidal function can replace another one in the summation. In our context, a spectral or wavelet component is simply the crosscorrelation coefficient of a given basis function with seismic data. If we choose to amplify or mute a given component and reconstruct the data from new weighted sums, we obtain an altered filtered version of the original data.
In our example of Fourier analysis, because the basis functions or the selected sinusoidal functions are orthogonal to each other, mathematicians would term the orthogonal Fourier transform to be an orthogonal transform. Such orthogonal transforms provide a minimum number of computed components that represent the measured seismic data. Orthogonal implementations of wavelet transforms in particular are effective at data compression. Because many of the wavelet components are very small, often we can represent a trace consisting of a thousand or more seismic samples with only one-tenth as many wavelet components.
Running-w indow spectral-filtering and spectral-balancing techniques have been applied to seismic data at least since the latter half of the s. In those applications, each seismic trace is broken into a suite of shorter, overlapping traces centered about the output sample. Longer windows provide more-robust statistics, with typical windows being or ms in length, and with shorter windows of ms providing poorer results.
In part, the poor results are because of an assumption that the underlying reflectivity has a white spectrum. If the underlying reflectivity is not white, it would be distorted during the spectral-balancing step. Because of this focus on spectral balancing, spectral analysis of shorter windows was overlooked until the middle s. The impetus for using spectral analysis of shorter. In general, seismic interpreters are quite content with relative spectral measurements, such as the frequency at which tuning occurs, and do not require the absolute value of each frequency component desired by processors, which are needed to reconstruct original data.
Because we only wish to interpret spectral components rather than to filter components and efficiently reconstruct the data, we no longer are bound by orthogonal transforms. As an example, if we wish to analyze seismic data within a ms window, Nyquist'S sampling criterion states that we only need to decompose the data at Hz increments - say, at 10,20,30,40,50, and 60 Hz for band-limited seismic data with a highest-frequency contribution between 60 and 70 Hz and no amplitude at 0 Hz.
Knowledge of the spectral components at these seven frequencies completely and uniquely describes this time series. Fast Fourier transforms FFT provide a particularly effective means of generating such components. Unfortunately, the efficiency of FFT is so ingrained in processors' heads, they forget that instead they could use a simple slow Fourier transform - that is, they could simply crosscorrelate any sine and cosine with the data.
In that manner, we will use slow Fourier transforms more properly called discrete Fourier transforms to decompose the data into sinusoidal components separated by I or 2 Hz, thereby allowing the interpreter to inspect a more finely sampled spectrum for features of interest. Because the Hz frequency interval will do the job, the use of the finer frequency sampling l or 2 Hz is extra work tbat provides no additional information. Although in theory we could reconstruct any given spectral component at the desired I-Hz interval from the minimal number of components defined by Nyquist's criterion at Hz intervals , it is simpler computationally to calculate them directly at the desired I-Hz sampling.
Likewise, orthogonal wavelet-transform components used in data compression also can be oversampled, allowing an interpreter to inspect a more finely sampled spectrum for features of interest. We begin this chapter by reviewing basic concepts of seismic resolution and thin-bed tuning. Then we briefly review the properties of Fourier transforms, emphasizing the impact of the shape and size of the seismic analysis window.
That will give us a quantitative understanding of the similarities and differences between two different methods: short-window discrete Fourier transforms commonly called spectral decomposition and wavelet transforms called instantaneous spectral analysis. After defining the terms and establishing the theoretical basis, we provide a suite of examples that exhibit these types of analysis in terms of both.
We close the chapter with a brief overview of the recently introduced SPICE spectral imaging of correlative events algorithm, showing its relationship to both instantaneous-attribute analysis and wavelet-decomposition analysis. Seismic Resolution and Thin-bed Tuning To better understand the effect of bed thickness on thin-bed resolution, we return to the simple wedge model shown in Chapter 5, Figure I. Generally, the top and bottom reflections from a thin bed do not have the equal and opposite values routinely used in seismic modeling of a thin-bed response.
Castagna recently demonstrated that this overly simplified model, in which reflection coefficients have equal magnitude but opposite signs, is pathological and that for the more general case the limits to vertical resolution are less severe. Accounting for attenuation also may allow us to increase the resolution Goloshubin et al. Nevertheless, we will follow Widess , Kallweit and Wood , and Robertson and Nogami and use this wedge model to understand why we can detect thin-bed anomalies even for this worst-case model.
The maximum constructive interference occurs when the wedge thickness is one-quarter of the effective-source wavelength or, when measured in two-way traveltime, it is one-half the thickness of the dominant period indicated by arrows in Figures Ic and l d of Chapter 5. For thicknesses smaller than that, the waveform stabilizes first and then rernains constant, and only the seismic amplitude changes with thickness.
Thus, below tuning, the frequency spectrum's shape does not change with changes in thickness because the waveform does not change shape. For this worst-case model which may not be valid in actual practice , Widess showed that when we are well below the one-quarter-wavelength tuning thickness, the amplitude changes linearly with thickness Chapter 5, Figure 2.
Kallweit and Wood examined this problem of resolution with a model consisting of two reflectors that have equal reflection coefficients of the same sign. The authors showed that when the thickness falls below that given by Rayleigh's criteria Figure I , it cannot be estimated from seismic data alone.
Kallweit and Wood's paper had a profound impact on the seismic processing community. Although the authors took care to point out the difference between detection of relative changes in thickness and resolution determination of a given thickness, a whole generation of seismic processors including the second author of this book apparently confused the issue and felt. Not until the s did Partyka and his colleagues revisit Widess's observations and show that amplitude variation with frequency, or more appropriately, amplitude variation across strata of varying thickness for a fixed frequency, could be used as a powerful interpretation tool.
Relevant Concepts of Fourier Analysis Fourier analysis is simply crosscorrelation of the seismic data with a suite of sines and cosines at predetermined frequencies. Each crosscorrelation coefficient between a given sine and cosine pair and the data is called a frequency component. Commonly, we use Euler's theorem:. We then express the crosscorrelation coefficients of sines and cosines with the data as a complex number, A w : A w. Regardless of the basis functions used, the transform coefficients are obtained by simply crosscorrelating each basis function with a window of the data in a fashion analogous to obtaining the Fourier coefficients through equation 6.
In this chapter, we limit ourselves to basis functions that are windowed sines and cosines and we use either the short-window discrete Fourier transform SWDFT or the wavelet-transform basis functions. They differ from each other in the application of their respective tapers.
For wavelet transforms, the tapering windows are proportional to the frequency of the sines and cosines and are shorter for higher frequencies. However, with the use of these new, tapered-windowed basis functions, we will observe nonzero amplitudes over a range of frequencies.
This smearing in the frequency domain occurs because the basis functions are tapered.
Thus, we need to understand the spectrum of the tapered window itself. It also is common to define A w in terms of its amplitude, a w , and its phase, J w : A w. Until now, research efforts have focused more on the amplitude component of the spectrum, a w. However, we expect that changes with respect to frequency of the phase, J w , may allow us to differentiate between upward-fining and upward-coarsening sequences, both of which have the same amplitude spectrum.
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Although sines and cosines are used for spectral decomposition, other transforms also can be used to decompose waveforms for interpretation. One major software vendor decomposes seismic data using orthogonal Tchebychev polynomials as a basis function, and another vendor derives nonorthogonal basis functions defined by the data themselves using self-organized maps, or alternatively, using. Figure 1. The definition of resolution between two reflectors with the same reflection coefficient that also has the same sign indicated by the vertical lines. The effective frequency of the source wavelet is defined as lib, where b is the wavelet breadth.
We can interpret the separation as being either in time for this chapter on thin-bed resolution or in space for later chapters on lateral resolution. Rayleigh's limit of resolution occurs when images are separated by the peakto-trough time interval, whereas Ricker's limit occurs when images are separated by a time interval equal to the separation between inflection points. After Kallweit and Wood Note that both the temporal window and the spectral width are of fixed size for all frequencies.
The result of higher or lower frequencies is simply to shift the analyzed spectra up or down. In contrast, the window functions wet that are used in wavelet-transform decomposition are typically Gaussian and have the form w kAl. By tapering the sharp corners of the analysis window, we minimize undesired side lobes in the frequency spectrum Figure 2b.
The same slab of data of fixed length 50 or ms will be analyzed by each frequency. We plot representative Morlet wavelets and their corresponding spectra in Figure 3. Note that in contrast to the SWDFT, the higher-frequency Morlet wavelets have shorter temporal extent but broader frequency spectra than the a I. Window width is a constant 0. Solid and dashed lines indicate the cosine and sine wavelets, respectively. Note the side lobes of the spectra.
AIso note that the spcctru m of the Hz wavelet extends into negative frequencies. Morlet wavelets commonly are used in wavelet compression and in instantaneous spectral analysis, using the window defined by w k6. Note that the bandwidth increases with center frequency.
Equally important, two Morlet wavelets of different frequency at the same analysis point will analyze different windows of seismic data.
Thus, if an interpreter wishes to explicitly analyze the frequency content of a specific fixedtemporal-window geologic interval across multiple frequencies, the SWDFT is more appropriate for the job. Looking ahead to the end of this chapter, we note that the recently introduced SPICE algorithm spectral imaging of correlative events uses a similar but significantly longer analysis window, a 3 'c, to provide greater frequency resolution at the expense of reduced temporal resolution Liner et al.
Note that although tile wavelets are longer in time, the spectra are narrower than those of the Morlet wavelet displayed in Figure 3. Although the window function, wet , is different for each of tile SWDFT, Morlet, and SPICE algorithms, they all are implemented by crosscorrelating the data, u t , with windowed sines and cosines, w t - i exp[jw t - i ], or equivalently, by crosscorrelating sines and cosines, exp[iw t - i ], with the windowed data, wet - r u t :.
In Figure 5, we show a reflectivity sequence. For simplicity, we assume that the spectra of the earth's reflectivity are white, implying that there is an equal probability of a given amplitude reflection coefficient anywhere within the time series. In general, we do not know the source wavelet. Instead, we process our data to look as though they have been acquired using a band-limited white source wavelet. One way of achieving such a balance is to crosscorrelate the seismic data, u t , with sines and cosines, thereby generating Fourier components.
Often, we compute an average spectrum that is representative of all the traces in the survey. We calculate the peak spectral amplitude, 0max' of that. Examining Figures 2 and 3, we observe the following. First, as defined, the effective data analysis window for the SWDFT is fixed whereas that for the wavelet transform is variable - shorter for higher frequencies and longer for lower frequencies. Second, the spectra of both basis functions are centered about the center frequency,! Third, the bandwidths of the tapered SWDFT basis functions are independent of frequency, whereas those of the wavelet transform are narrower at low-center frequencies and broader at highcenter frequencies.
Most important, all spectral components, whether we are using the short-window DFT or the variablewindow CWT continuous-wavelet transform , have contributions from neighboring frequencies. Because most interpreters are not experienced seismic data processors, we emphasize that although we may see geologic features of interest at a 2-Hz component, rarely do we actually record such low-frequency data.
Instead, we display the information content from higher frequencies say, Hz that fall within the tails of the frequency spectra displayed in Figures 2b, 3b, and 4b. Such a pitfall is particularly prevalent when we use an insufficiently tapered SWDFT, in which side lobes in the frequency spectrum can mix in even more frequencies.
The seismic wavelet for 10 Hz goes beyond the scale of this plot. We then calculate a spectral compensation factor 1. If this compensation factor is applied to the average or median spectra, it generates a result comparable to that displayed in Figure 6b. When this compensation factor is applied to the spectra of the individual windowed traces, it statistically removes the effect of the unknown source wavelet, thereby resulting in a colored spectrum that is representative of the colored reflectivity within the analysis window.
Finally, we or our software reload the spectrally balanced frequency slices into the interpretation workstation for analysis Figure 8d. The spectral decomposition SWDFT frequency slices allow the interpreter to visualize interference patterns, such as thin-bed tuning associated with channels and deltas in plan view. This is not a magical process creating information out of nothing. The SWDFT has taken time slices and combined them with variable weights the sines and cosines used in crosscorrelation to form frequency slices.
The interpreter animates through these images and chooses the images that fit his or her geologic model. Because the basis functions are not orthogonal, many of the images are redundant, with much similarity between images created at neighboring frequencies. Other images contain only noise. That is the case for images at frequencies at tile low and high ends of the seismic source's frequency spectrum. As the following examples show, identification of textures and patterns that are indicative of geologic processes is proportional to the interpreter's skills and understanding of the depositional environment.
If he lacks 4D interpretation software, the interpreter simply loads these volumes into tile workstation in multiplexed form. Examples Our first example comes from Partyka et al. In Figure 9 we display a conventional amplitude extraction Figure 9a and an instantaneous envelope extraction Figure 9b along a Pleistocene-age horizon from. Figure 6. The concept of spectral balancing to achieve a band-limited white spectrum in the presence of noise.
South Marsh Island, on the continental shelf in the Gulf of Mexico. These images are of the paleo-Mississippi River, approximately km west of the river's current position. We see a complex distributary system, including bifurcating channels, point bars, and longitudinal bars that are controlled by faults in the southeastern portion of the image. The narrow channel, A, is poorly imaged in Figure 9 but is clearly visible on tile Hz spectral component in Figure lOb.
The same channel appears fainter on tile Hz image in Figure lOa, implying that 26 Hz is closer to the tuning frequency than 16 Hz is. In contrast, although channel B appears in all four images in Figures 9 and 10, it has maximum lateral resolution in the Hz spectral component Figure l Oa , in which we sec discrete meander loops indicated by the white arrow. The wider channel, B, is better imaged tuned in the lower-frequency, Hz image, whereas the narrow channel, A, is better imaged tuned at the higher, Hz frequency.
That finding is consistent with the well-established correlation between channel width and thickness. Laughlin et al. Figure 7. Short-window spectral decomposition and the convolutional model. Although we may wish to assume that the reflectivity has a more or less white spectrum, any short-windowed realization of this white distribution will have only a few discrete reflection spikes and thus invariably will have a colored spectrum.
If we process the data to provide a band-Jimited white source spectrum and assume we have white noise, the colored spectrum of the windowed seismic data will be a band-limited representation of the colored spectrum of the reflectivity within the window. After Partyka et al, The above-described model is illustrated clearly in our second example, taken from a 3D onshore data volume over a real channel Figure 12 Bahorich et aJ.
Although an amplitude map showed reasonable detail about the shape of the reservoir, the spectral-decomposition images clearly illuminated the thickest and thinnest sequences in the reservoir. Amplitude maps of certain frequencies showed the thinning of levies - information that helped the interpreter map the detailed geometry of the reservoir. That geometry was confirmed subsequently by well control.
Our third example comes from Peyton et al. The authors merged three different 3D surveys into a single survey covering the area of study Figure Their objective was to map multiple stages of incised valleys into three coarsening-upward marine parasequences the lower, middle, and upper Red Fork sandstone bounded by the region-. Band-limited colored spectrum ally extensive Pink limestone above and the Inola Limestone Member of the Boggy Shale and me Novi limestone below Figure Variable sediment fill in the incised channels results in a complex internal architecture that is difficult to interpret on conventional horizon slices through the seismic data.
This study focuses on the upper Red Fork incised-valley system, which is the largest such system and which images most clearly on 3D seismic and also contains the best reservoir rocks in the area. Before acquisition of the 3D surveys, it was believed that the valley fills in this region occurred in four stages, with stage III being the most abundant hydrocarbon producer.
Of the several wells drilled, some were believed to have penetrated the edge of stage III valley fill; however, those wells did not produce. Wells producing from stage III sands indicate a gap in the production shown in yellow in Figures 14b and 15b. The fairly large width 0. Thus, the area was covered with 3D seismic surveys in an attempt to reduce risk and to explore undrilled potential in stage III sands. The seismic data. This new scaled spectrum is equivalent to the spectrum of the idealized source wavelet shown in Figure 5. Deconvolution algorithms are designed to achieve a similar result - to generate a band-limited processed-data spectrum by eliminating multiples and reverberations that otherwise would color it.
Although we may still wish to assume that the reflectivity is white, tbe fact that we have only samples in our lOO-ms analysis window implies that we have only a specific realization drawn from a white spectrum. In general, the spectrum of that windowed reflectivity series is not white, it is colored. When the resulting data window, lI f , is multiplied by the white spectrum of the source wavelet, it also has a colored spectrum. We address how to obtain a white source-wavelet spectrum from windowed data in the next section.
But if we can, we want the spectrum of the seismic data to be a band-limited version of the spectrum of the reflectivity series within the analysis window. Figure 5. Long-window spectral decomposition and the convolutional model. Typically, we assume that the multiple-free reflectivity spectrum has very little structure in this case, we display a white spectrum.
Next, we calculate the Fourier spectrum of the entire unwindowed trace. Finally, we explicitly flatten its spectrum, which, under the assumption of white additive noise, will flatten the spectrum of the source wavelet. Similar assumptions are made in the deconvolution algorithms routinely used in seismic processing. After Partyka et al. We select the zone of interest, which typically follows an interpreted horizon. We select the horizon and define a constant-thickness slab of data that lie a given number of milliseconds above and below the selected horizon in the seismic data volume Figure 8a.
We or our software apply a OFT to the slab of data, frequency by frequency, generating a sequence of constant-frequency spectral-component maps Figure 8c. Next, we assume that the geology and therefore reflectivity are sufficiently random across the entire zone of interest for the average reflectivity spectrum to be white. Source wavelet. In between these two markers, the incised valleys are characterized by discontinuous reflections of varying amplitudes. Obviously, interpreters will find it difficult to use traditional interpretation techniques e. Furthermore, it is difficult to identify the individual stages of fill.
As was eventually learned, cross section AA' shown in Figures begins in the regional Red Pork marine parasequences in the south, cuts through a. J d Finally, we spectrally balance the results as defined in Pigurc 6, and interpret spectral components through animation, composite displays as in Figures 18, 19, and 22a , 30 visualization as in Figure 21 , or statistical analysis as in Figures 20 and 22b.
After Johann et a. Figure 9. Unlike the simple amplitude extraction shown in our earlier Gulf of Mexico example in Figure 9a, reflectivity over the Red Fork incised valley changes significantly with the valley fill, making horizon slices through the seismic data diff-icult to interpret. For that reason, Peyton et al. Because the Red Fork interval was ms thick Figure 14 , spectral components ranging from 20 Hz to 50 Hz were computed at I-Hz intervals within a ms window parallel to, but not including, the Novi limestone.
In Figure 15a, we show the Hz amplitude slice from the. Spectral decoma. That sequence was not readily seen on.
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A Hz amplitude slice from spectral decomposition of Red Fork deposits a without and b with current inter-. After Peyton et al. The cross-section datum is the top of the Novi limestone. Upper Red Fork valley stages and regionally correlative limestones are shaded. The Griffin I well perforations shown has produced 1. After Peyton et a1. The black arrow indicates an amplitude anomaly. The dashed blue line indicates a fault. After Baboricb et a1. Whereas an orthogonal CWT provides the most efficient means of representing seismic data with the fewest number of wavelet components, the matched-pursuit technique is designed to provide wavelet components of greater interest to the interpreter.
Liu and Marfurt S offered details of such an algorithrn based on Ricker and Morlet wavelets using the flow shown in Figure They began by precomputing a table of complex wavelets for a finely sampled suite of frequencies. The real and imaginary components of those wavelets are simply the cosine and sine wavelets shown in Figure 3a.
Next, they generated a complex trace using Hilbert transforms, from which they calculated the instantaneous envelope and frequency of each trace. They then searched for the largest values of the envelope and its corresponding instantaneous frequency. Next, they least-squares-fitted that suite of wavelets to the complex trace, solving for complex coefficients that correspond to the amplitude and phase of the complex wavelet that best fits the data.
Finally, they subtracted that amplitude and phase-rotated complex wavelet from the current version of the data, thereby generating a new residual trace. They repeated the process on the residual until the energy of the residual trace fell below a user-defined threshold. Liu and Marfurt S illustrate this process through the example shown in Figure The result after the first iteration is shown in the left column, the result after the fourth iteration is in the center column, and the result after the sixteenth iteration is in the right column.
The location and magnitude of the wavelet envelope are shown in Figure 24d. Each wavelet envelope, frequency, and phase is extracted from the precomputed complex wavelet table and added in to generate the modeled seismic data Figure 24a. These modeled seismic data are subtracted from the original seismic data to generate residual seismic data Figure 24b.