Stratigraphic coordinates, a coordinate system tailored to seismic interpretation

Application of stratigraphic coordinates to spectral decomposition

Improving the accuracy of spectral decomposition is one of the possible applications of the stratigraphic coordinate system. Spectral decomposition is a window-based analysis to characterize the reflecting wavelet of an interpretation target and refers to any method that produces a continuous time-frequency analysis of a seismic trace (Partyka and Lopez, 1999). According to the convolutional model, seismic traces are considered as normal-incidence 1D seismograms, which is true in the case of horizontal layers and allows for capturing the signal wavelet while performing spectral decomposition along the seismic trace. However, when the subsurface exhibits dipping layers, the convolutional model no longer holds true, and sampling the seismic waveform vertically instead of perpendicularly to reflectors introduces a dip-dependent stretch that will carry over to any frequency estimation or spectral decomposition. Guo and Marfurt (2010) proposed to solve this problem by sampling the signal wavelet along the ray-path on which the wavelet travels. Because this path is normal to reflectors, we implement the same idea by employing the stratigraphic coordinate system, which honors the convolutional model and can capture and analyze seismic waveforms perpendicularly to seismic reflectors. Figure 8 shows Gulf of Mexico data reproduced from Lomask et al. (2006) and Liu et al. (2011) that contain a salt dome and horizons that dip steeply on the flank of the dome because of the salt piercement. Following Liu et al. (2011) we calculated the spectral decomposition of the data in the Cartesian coordinate system. Figure 9 shows horizon slices from spectral decomposition calculated in the Cartesian coordinate system at different frequencies because depositional elements of different thicknesses tune it at different frequencies. Figure 10 also shows the same horizon slices as Figure 9, but this time from spectral decomposition calculated in the stratigraphic coordinate system. Compared with horizon slices in Figure 9, those from spectral decomposition calculated in the stratigraphic coordinate system better highlight detailed geologic features such as sand channels. That is because in the stratigraphic coordinates seismic horizons get flattened and the vertical direction corresponds to the normal direction to reflectors. We can therefore analyze the unbiased seismic wavefrom and achieve a more accurate spectral decomposition result. Indeed, since seismic reflectors appear flat in the stratigraphic coordinate system, (vertical) trace analysis methods such as spectral decomposition probe the unbiased seismic waveform and thus yield more accurate measurements and attributes. Conversely, methods such as post-stack seismic inversion, spiking deconvolution, tuning analysis, etc., usually assume that layers are flat and might therefore lead interpreters to incur errors in the presence of dips. The same methods, just like spectral decomposition, may benefit from being applied in the stratigraphic coordinate system and thereby produce results unbiased by structural dip.

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Figure 8. Seismic image from Gulf of Mexico. (a) Time slice. (b) Inline section. (c) Cross-line section.

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Figure 9. Horizon slices from spectral decomposition at (a) 10 Hz, (b) 20 Hz, (c) 30 Hz, and (d) 40 Hz in Cartesian coordinate system.

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Figure 10. Horizon slices from spectral decomposition at (a) 10 Hz, (b) 20 Hz, (c) 30 Hz, and (d) 40 Hz in stratigraphic coordinate system. The 30 Hz slice most clearly displays visible channel features (red arrows).

Stratigraphic coordinates, a coordinate system tailored to seismic interpretation