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**Interpretational Applications of Spectral Decomposition**

Greg Partyka, James Gridley, and John Lopez

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**Spectral Decomposition**

uses the discrete Fourier transform to: quantify thin-bed interference, and detect subtle discontinuities.

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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Long Window Analysis The geology is unpredictable. Its reflectivity spectrum is therefore white/blue.

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**Long Window Analysis TIME DOMAIN FREQUENCY Reflectivity r(t)**

Fourier Transform Amplitude Frequency Wavelet w(t) Noise n(t) Seismic Trace s(t) TIME DOMAIN FREQUENCY Travel Time

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Short Window Analysis The non-random geology locally filters the reflecting wavelet. Its non-white reflectivity spectrum represents the interference pattern within the short analysis window.

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**Short Window Analysis TIME DOMAIN FREQUENCY Reflectivity r(t)**

Wavelet Overprint Reflectivity r(t) Fourier Transform Amplitude Frequency w(t) Noise n(t) Seismic Trace s(t) TIME DOMAIN FREQUENCY Travel Time

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**Spectral Interference**

The spectral interference pattern is imposed by the distribution of acoustic properties within the short analysis window.

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**Spectral Interference**

Source Wavelet Amplitude Spectrum Thin Bed Reflection Thin Bed Reflection Reflected Wavelets Source Wavelet Reflectivity Acoustic Impedance Temporal Thickness Fourier Transform Amplitude Frequency 1

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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**Wedge Model Response REFLECTIVITY FILTERED (Ormsby 8-10-40-50 Hz)**

Temporal Thickness (ms) REFLECTIVITY FILTERED (Ormsby Hz) SPECTRAL AMPLITUDES 10 20 30 40 50 100 200 Travel Time (ms) Frequency (Hz) Temporal Thickness 1 0.0015 Amplitude Amplitude spectrum of 10ms blocky bed Amplitude spectrum of 50ms blocky bed 10Hz spectral amplitude 50Hz spectral amplitude

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**Individual Amplitude Spectra**

The temporal thickness of the wedge (t) determines the period of notching in the amplitude spectrum (Pf) with respect to frequency Amplitude spectrum of 10ms blocky bed. Amplitude spectrum of 50ms blocky bed. Pf = 1/t where: Pf = Period of amplitude spectrum notching with respect to frequency. t = Thin bed thickness. 20 40 60 80 100 120 140 160 180 200 220 240 0.0002 0.0004 0.0006 0.0008 0.0010 0.0014 0.0012 Frequency (Hz) Amplitude

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**Wedge Model Response REFLECTIVITY FILTERED (Ormsby 8-10-40-50 Hz)**

Temporal Thickness (ms) REFLECTIVITY FILTERED (Ormsby Hz) SPECTRAL AMPLITUDES 10 20 30 40 50 100 200 Travel Time (ms) Frequency (Hz) Temporal Thickness 1 0.0015 Amplitude Amplitude spectrum of 10ms blocky bed Amplitude spectrum of 50ms blocky bed 10Hz spectral amplitude 50Hz spectral amplitude

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**Discrete Frequency Components**

The value of the frequency component (f) determines the period of notching in the amplitude spectrum (Pt) with respect to bed thickness. 10Hz spectral amplitude. 50Hz spectral amplitude. 10 20 30 40 50 0.0002 0.0004 0.0006 0.0008 0.0010 0.0014 0.0012 Amplitude Temporal Thickness (ms) Pt = 1/f where: Pt= Period of amplitude spectrum notching with respect to bed thickness. f = Discrete Fourier frequency.

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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**The Tuning Cube 3-D Seismic Volume Interpret Interpreted Subset**

x y z freq Interpret 3-D Seismic Volume Subset Compute Animate Interpreted Zone-of-Interest Subvolume Tuning Cube (cross-section view) Frequency Slices through Tuning Cube (plan view)

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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**Prior to Spectral Balancing**

The Tuning Cube contains three main components: thin bed interference, the seismic wavelet, and random noise Multiply Tuning Cube x y freq Seismic Wavelet Noise Thin Bed Interference + Add

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**Short Window Analysis TIME DOMAIN FREQUENCY Reflectivity r(t)**

Wavelet Overprint Reflectivity r(t) Fourier Transform Amplitude Frequency w(t) Noise n(t) Seismic Trace s(t) TIME DOMAIN FREQUENCY Travel Time

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**Spectral Balancing Split Spectral Tuning Cube**

x y freq Split Spectral Tuning Cube into Discrete Frequencies Tuning Cube Spectrally Balanced Gather Discrete Frequencies into Tuning Cube Independently Normalize Each Frequency Map Frequency 1 Frequency 2 Frequency 3 Frequency 4 Frequency n Frequency Slices through Tuning Cube (plan view)

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**After Spectral Balancing**

The Tuning Cube contains two main components: thin bed interference, and random noise Tuning Cube x y freq Noise Thin Bed Interference + Add

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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Real Data Example Gulf-of-Mexico, Pleistocene-age equivalent of the modern-day Mississippi River Delta.

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**N Response Amplitude Channel “A” Fault-Controlled Channel Point Bar**

Gulf of Mexico Example 10,000 ft Channel “A” Channel “B” Fault-Controlled Channel Point Bar N 1 Amplitude analysis window length = 100ms

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**N Tuning Cube, Amplitude at Frequency = 16 hz Channel “A”**

Gulf of Mexico Example 10,000 ft North-South Extent of Channel “A” Delineation Channel “A” Channel “B” Fault-Controlled Channel Point Bar N 1 Amplitude analysis window length = 100ms

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**N Tuning Cube, Amplitude at Frequency = 26 hz Channel “A”**

Gulf of Mexico Example 10,000 ft North-South Extent of Channel “A” Delineation Channel “A” Channel “B” Fault-Controlled Channel Point Bar N 1 Amplitude analysis window length = 100ms

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**Hey…what about the phase?**

Amplitude spectra delineate thin bed variability via spectral notching. Phase spectra delineate lateral discontinuities via phase instability. Phase Spectrum Phase Frequency Amplitude Spectrum Amplitude Thin Bed Reflection Fourier Transform

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**Faults N Response Phase Gulf of Mexico Example 10,000 ft Phase 180**

-180 Phase Gulf of Mexico Example

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**Faults N Tuning Cube, Phase at Frequency = 16 hz**

10,000 ft N 180 -180 Phase analysis window length = 100ms Gulf of Mexico Example

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**Faults N Tuning Cube, Phase at Frequency = 26 hz**

analysis window length = 100ms Faults 10,000 ft N 180 -180 Phase Gulf of Mexico Example

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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**Discrete Frequency Energy Cubes**

Compute 3-D Seismic Volume x y freq z z = 1 z = n z = 3 z = 4 z = 5 z = 6 z = 2 Subset Time-Frequency 4-D Cube Discrete Frequency Energy Cubes Frequency 1 Frequency 2 Frequency 3 Frequency 4 Frequency m

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Outline Convolutional Model Implications Wedge Model Response The Tuning Cube Spectral Balancing Real Data Examples Alternatives to the Tuning Cube Summary

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Summary Spectral decomposition uses the discrete Fourier transform to quantify thin-bed interference and detect subtle discontinuities. For reservoir characterization, our most common approach to viewing and analyzing spectral decompositions is via the “Zone-of-Interest Tuning Cube”. Spectral balancing removes the wavelet overprint. The amplitude component excels at quantifying thickness variability and detecting lateral discontinuities. The phase component detects lateral discontinuities.

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