# Spectral decomposition

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Figure 1: This image is a cartoon cross section showing the frequency-temporal thickness relationship represented by spectral decomposition. The green color shows areas tuned at 30 Hz and the red represents tuning at 18 Hz. After Laughlin et al. (2002)

Spectral Decomposition or time-frequency analysis (also time-frequency decomposition) is a method employed to aid in the interpretation of seismic data. Spectral decomposition can be performed on a multitude of attributes (frequency, dip, azimuth…), though the frequency is the most common. It can also be performed on either time migrated or depth migrated data and results in tuning frequencies with units of Hz and cycles/distance, respectively. The result of spectrally decomposing data is the frequency and phase components of which the former is a direct measure of the relative seismic amplitude within a frequency band [1] [2]. The main usage of the attribute is to help with stratigraphic interpretation by improving thin bed resolution and showing temporal bed thickness variability. This is necessary because a wavelet will often span multiple subsurface layers, resulting in a complex tuned reflection which is a result of the convolutional model. For this reason, spectral decomposition is said to remove the wavelet overprint from seismic data [3]. Figure 1 to the left is a good representation of the ultimate result of spectral decomposition (thickness variation within a stratigraphic body). The first of the two discrete frequencies is 30 Hz (green) and illuminates the thinnest part of the channel in map view. In red, 18 Hz is shown and shows the thicker parts of the channel (thalwegs).

## Methods

Though overprinted by the wavelet in the recorded seismic data, thin beds still possess unique frequency expressions. Thus, if the recorded traces are transformed from the time (or depth) to frequency domain (Fourier Analysis), the spectra can be extracted. There are numerous methods to perform this transformation and create spectral decomposition time slices and volumes (listed below), all of which are computed on a trace by trace basis [4]. One of these methods is termed a Short-Window Discrete Fourier Transform (SWDFT). In essence, this method takes vertical slices, or entire volumes and assigns weight to traces (or cross-correlation with tapered sines and cosines) over a fixed time window [1]. This time window is centered about each analysis point at a given time. Commonly, window lengths range from 50 – 100 ms with a taper of around 20 % of the window length [1]. Continuous-Wavelet Transform (CWT) is another common method utilized in spectral decomposition. It is similar to the SWDFT, however, CWT has a varying window length (taper) which is proportional to the central frequency, being shorter for higher frequencies and longer for lower frequencies [1]. The result of these differences in window length is in resolution. The SWDFT will have a fixed time-frequency resolution, while the CWT will not [5].

### Methods for Spectral Decomposition:

• Fourier Transform
• Short-time Discrete Fourier Transform
• S transform or Stockwell Transform
• Continuous-Wavelet Transform
• Matching pursuit and other dictionary-based transforms
• Empirical Mode Decomposition
• Wigner-Ville distribution
• Synchrosqueezing transform

## Visualization

Figure 2: The image shows three individual spectral components at a) 30 Hz b) 40 Hz c) 50 Hz with gradational color schemes and two blended images d) RGB Blend (red = 30 Hz, green = 40 Hz, Blue = 50 Hz) and e) HLS blended. After Hall and Trouillot (2004).

Individual spectral decomposition frequency slices or volumes (at a single frequency) can be displayed in a variety of software and with a multitude of color schemes. However, there are two often used color options which effectively blend multiple spectrally decomposed images. These are RGB and hue-lightness-saturation HLS). With RGB, the user selects three discrete frequencies and plots them against red, green, and blue. HLS stands for hue (color wavelength), lightness (color brightness), and saturation (color tinting) [[1]]. In this display, three generated amplitude slices are represented by these three quantities and then combined to give a single image [1]. A representation of these color schemes is shown in Figure 2. In the figure, a, b, and c are individual spectral components with white being bright responses. Part d is an RGB blend of the three frequency slices and e is an HLS image in which bright red shows the locations where all three frequencies are bright [1]. Figure 3 is a higher resolution RGB spectral decomposition image which reveals one large meandering river system and a few smaller, less sinuous channels. The thickest channels appear as orange, the green/yellow locations are less thick, and the blue areas are thin (mostly muddy overbank deposits).

Figure 3: The above image is an RGB blend in a stratigraphically complex reservoir off the coast of West Africa. High Frequency - Blue, Middle Frequency - Green, Low Frequency - Red. After Bahorich et al. (2002).

## Application

Spectral Decomposition has become widely used in the Petroleum industry as a part of geophysical workflows for the interpretation of 3-D seismic data. In combination with variance (a.k.a. semblance, coherence) channels can be more easily identified and analyzed. Coherence illuminates the channel edges while spectral decomposition represents the channel thickness. Additionally, this method can provide a better idea of channel body continuity, fill variability, and possible reservoir quality [6].

Apart from the traditional uses of spectral decomposition in the form of frequency slices, it can be combined with traditional analysis of amplitude variation with offset to calculate spectral AVO. In this method seismic gathers are spectrally decomposed to various frequencies of interest and then AVO is performed. The advantage of this method is that stacked thin gas sands which are normally not seen in traditional AVO can be easily predicted by this method [7].

Another application is the incorporation of spectral decomposition into other seismic attributes such as SPICE or Voice Components. SPICE (spectral imaging of correlative events) is an attribute which enhances the convergence of waveforms. This method uses the coefficients of wavelet transforms and the Holder exponent, which is a measure of a function’s singularity [1]. The result is a volume which accentuates discontinuities and allows for autopickers to track horizons in stratigraphically complex areas. An example of SPICE is shown in figure 4 which displays its effectiveness at singularity evaluation. In the amplitude data (figure 4a) there is a convergence of reflectors which are difficult to separate. However, in part b) the SPICE attribute shows more clearly shows the discontinuity between the reflectors. Voice Components is a function of spectral magnitude and phase at each time-frequency sample taken from spectral decomposition. The attribute itself is a band-pass filtered version of the input seismic data which typically illuminates discontinuities [8]

Figure 4: The images above show a) a vertical slice through amplitude data and b) a vertical slice through a SPICE volume. After Liner et al. (2004).

## Issues

One issue which has become increasingly common in the interpretation of seismic data is that of overwhelming data (specifically attributes). Spectral decomposition is a large contributor to this problem. Often there are numerous phase and spectral magnitude volumes or slices generated, depending on the frequency increment selected [2]. A remedy to this issue is a dimensionality reduction via principal component analysis which analyzes attributes and yields a few volumes representing the maximum variation of the input attributes[9]

Another issue is with visual display. As mentioned previously, there are methods for blending discrete frequencies for more coherent interpretation. However, the frequency spectra are highly variable and thus three components may be insufficient in detail. One method to resolve this is optical stacking of a larger range of frequencies [10].

## Examples

### Interpretation of Incised Valley Fill

A widely known spectral decomposition study was performed by Peyton et al. in 1999. In their research, the group used a combination of spectral decomposition and coherence to delineate and interpret incised valleys of the Upper Red Fork interval in the northern Anadarko basin. Spectral decomposition and coherence were applied to an Amoco mega merge 3D seismic survey. The goal was to identify the discontinuous channel bodies and possibly interpret the stages of channel fill. Using the two attributes, the group was able to achieve both. In Figure 5, one result of the study is shown which displays a spectral component slice at a frequency of 36 Hz. Part 5a shows the channel delineated by coherence, as well as the amplitude differences within its bounds. Using knowledge of the Upper Red Fork Interval and well logs (part 5b) they were able to interpret the different stages of valley fill along the entire channel [11].

Figure 5: The above image shows a spectral component slice at 36 Hz with a) no interpretation b) interpreted valley fill stages. After Peyton et al. (1999).

### Interpretation of Channel Continuity

A more recent example is that of Othman et al. 2016 in which spectral decomposition was utilized to map bed thickness, image geologic discontinuities and delineate channels in the Western Desert Deep Marine oil province 110 km NE of Alexandria, Egypt[6]. Two channels (termed Red and Yellow) were discovered in the amplitude data, both of which dip to the north and pinch out laterally. The Red channel was easily identified and followed in the amplitude data with two clear internal discontinuities (labeled in figures 6 and 7). However, the Yellow channel was dimmer and presented difficulties in reflector tracking. As spectral decomposition has been proven to aid in channel interpretation, the choice was made to employ this technique. So, spectral decomposition volumes were generated around the interpreted horizons, and the various frequency volumes were interpreted. For the Red channel, the frequencies of 5, 25, and 65 Hz were chosen as they revealed different features of the channels. Figure 7 shows the result of an RGB blend performed with the three separated frequency volumes. The image displays the majority of the channel with bright coloration, meaning it contains all three frequencies and corresponds to the thickest portion of the channel. The two discontinuities seen as low amplitudes in figure 6 are seen in figure 7 as low-frequency zones (5 Hz) between the mixed frequency areas. This was interpreted as showing the channel was connected in these regions. Additionally, where the channel fairway was undefined in the amplitude maps, it can be clearly seen at 65 Hz in the frequency data (figure 7). A similar process was also performed for the Yellow channel which revealed a multi-story channel complex.

Figure 6: Average Absolute Amplitude attribute extracted onto the top Red channel surface. The channel is clearly shown by generally high average absolute amplitude. The image has been annotated to illuminate channel features. After Othman et al. (2016)
Figure 7: A blended image of spectral frequencies 5 Hz, 25 Hz, and 65 Hz extractracted onto the top Red Channel surface. After Othman et al. (2016)

## References

1. Chopra, S., & Marfurt, K. J. (2007). “Chapter 6: Spectral Decomposition and Wavelet Transforms” Seismic attributes for prospect identification and reservoir characterization (Vol. ISBN 978-1-56080-141-2, ISBN 978-0-931830-41-9). doi:https://doi.org/10.1190/1.9781560801900
2. Chopra, S., & Marfurt, K. J. (2015). Enhancing the interpretability of seismic data with spectral decomposition phase components. Reading presented at SEG Annual Meeting in the United States, New Orleans.
3. Partyka, G., Gridley, J., & Lopez, J. (1999). Interpretational applications of spectral decomposition in reservoir characterization. The Leading Edge, 18(3), 353-360. doi:10.1190/1.1438295
4. Tengfei Lin, Bo Zhang, Shiguang Guo, Kurt Marfurt, Zhonghong Wan, and Yi Guo (2013) Spectral decomposition of time- versus depth-migrated data. SEG Technical Program Expanded Abstracts 2013: pp. 1384-1388. https://doi.org/10.1190/segam2013-1166.1
5. Sinha, S., Routh, P. S., Anno, P. D., & Castagna, J. P. (2005). Spectral decomposition of seismic data with continuous-wavelet transform. Geophysics, 70(6), 19-25. doi:10.1190/1.2127113
6. , A. A., Fathy, M., & Maher, A. (2016). Use of spectral decomposition technique for delineation of channels at Solar gas discovery, offshore West Nile Delta, Egypt. Egyptian Journal of Petroleum, 25(1), 45-51. doi:10.1016/j.ejpe.2015.03.005
7. Saputro, J., J., Samudra, A. B., Lestari, E. P., S., Ramadhan, A., & Hirosiadi, Y. (2016). Combined AVO and Spectral Decomposition Analyses to Characterize Gas Sand Reservoir Below Tuning Thickness Condition. Proc. Indonesian Petrol. Assoc., 40th Ann. Conv. doi:10.29118/ipa.0.16.136.g
8. Chopra, S., & Marfurt, K. J. (2016). Spectral decomposition and spectral balancing of seismic data. The Leading Edge, 35(2), 176-179. doi:10.1190/tle35020176.1
9. Chopra, S., & Marfurt, K. J. (2014). Churning seismic attributes with principal component analysis. SEG Technical Program Expanded Abstracts 2014. doi:10.1190/segam2014-0235.1
10. Johann, P., G. Ragignin, and M. Spinola, 2003, Spectral decomposition reveals geological hidden features in the amplitude maps from a deep-water reservoir in the Campos Basin: 73rd Annual International Meeting, SEG, Expanded Abstracts, 1740–1743.
11. Peyton, L., Bottjer, R., & Partyka, G. (1998). Interpretation of incised valleys using new 3-D seismic techniques: A case history using spectral decomposition and coherency. The Leading Edge, 17(9), 1294-1298. doi:10.1190/1.1438127