Time-Frequency Representation of Microseismic Signals using the SST
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The document introduces the synchrosqueezing transform (SST) as a novel high-resolution tool for time-frequency analysis of microseismic signals, addressing limitations of classical methods like STFT and CWT. SST enables better signal reconstruction and extraction of individual components, with applications in frequency measurement, hydrocarbon detection, and signal enhancement. The paper concludes that SST is effective for high-resolution analysis and reconstruction of microseismic signals, making it valuable for various geophysical applications.
( F
t](https://image.slidesharecdn.com/csegssthenry2013-130929210307-phpapp01/85/Time-Frequency-Representation-of-Microseismic-Signals-using-the-SST-6-320.jpg)
![Synthetic Example 1
Noiseless synthetic signal 𝑠 𝑡 as the sum of the
following components:
𝑠1 𝑡 = 0.5 cos 10𝜋𝑡 , 𝑡 = 0: 6 𝑠
𝑠2 𝑡 = 0.8 cos 30𝜋𝑡 , 𝑡 = 0: 6 𝑠
𝑠3(𝑡) = 0.7 cos 20𝜋𝑡 + sin(𝜋𝑡) , 𝑡 = 6: 10.2 𝑠
𝑠4(𝑡) = 0.4 cos 66𝜋𝑡 + sin(4𝜋𝑡) , 𝑡 = 4: 7.8 𝑠
𝑓𝑠1 (𝑡) = 5
𝑓𝑠2 (𝑡) = 15
𝑓𝑠3 𝑡 = 10 + cos 𝜋𝑡 /2
𝑓𝑠4 𝑡 = 33 + 2 ∗ cos 4𝜋𝑡
𝑓 𝑡 =
𝑑𝜃(𝑡)
2𝜋 𝑑𝑡
0 2 4 6 8 10
Signal
Component 4
Component 3
Component 2
Component 1
Synthetic 1
Time [s]](https://image.slidesharecdn.com/csegssthenry2013-130929210307-phpapp01/85/Time-Frequency-Representation-of-Microseismic-Signals-using-the-SST-12-320.jpg)
![Real Example: TFR. 5 minutes of data
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-1
0
1
2
x 10
-3
Time [min]
Amplitude[Volts]](https://image.slidesharecdn.com/csegssthenry2013-130929210307-phpapp01/85/Time-Frequency-Representation-of-Microseismic-Signals-using-the-SST-18-320.jpg)
Time-Frequency Representation of Microseismic Signals using the SST
- 1. Time-Frequency Representation of Microseismic Signals using the Synchrosqueezing Transform Roberto H. Herrera, J.B. Tary and M. van der Baan University of Alberta, Canada rhherrer@ualberta.ca
- 2. New Time-Frequency Tool • Research objective: – Introduce two novel high-resolution approaches for time-frequency analysis. • Better Time & Frequency Representation. • Allow signal reconstruction from individual components. • Possible applications: – Instantaneous frequency and modal reconstruction. – Multimodal signal analysis. – Nonstationary signal analysis. • Main problem – All classical methods show some spectral smearing (STFT, CWT). – EMD allows for high res T-F analysis. – But Empirical means lack of Math background. • Value proposition – Strong tool for spectral decomposition and denoising . One more thing … It allows mode reconstruction.
- 3. Why are we going to the T-F domain? • Study changes of frequency content of a signal with time. – Useful for: • attenuation measurement (Reine et al., 2009) • direct hydrocarbon detection (Castagna et al., 2003) • stratigraphic mapping (ex. detecting channel structures) (Partyka et al., 1998). • Microseismic events detection (Das and Zoback, 2011) • Extract sub features in seismic signals – reconstruct band‐limited seismic signals from an improved spectrum. – improve signal-to-noise ratio of the attributes (Steeghs and Drijkoningen, 2001). – identify resonance frequencies (microseismicity). (Tary & van der Baan, 2012).
- 4. The Heisenberg Box FT Frequency Domain STFT Spectrogram (Naive TFR) CWT Scalogram 𝜎𝑡 𝜎𝑓 ≥ 1 4𝜋 Time Domain All of them share the same limitation: The resolution is limited by the Heisenberg Uncertainty principle! There is a trade-off between frequency and time resolutions The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. --Heisenberg, uncertainty paper, 1927 http://www.aip.org/history/heisenberg/p08.htm Hall, M. (2006), first break, 24, 43–47. Gabor Uncertainty Principle
- 5. • Localization: How well two spikes in time can be separated from each other in the transform domain. (Axial Resolution) • Frequency resolution: How well two spectral components can be separated in the frequency domain. 𝜎𝑡 = ± 5 ms 𝜎𝑓= 1/(4𝜋*5ms) ± 16 Hz Hall, M. (2006), first break, 24, 43–47. 𝜎𝑡 𝜎𝑓 ≥ 1 4𝜋 The Heisenberg Box
- 6. Heisenberg Uncertainty Principle Choice of analysis window: Narrow window good time resolution Wide window (narrow band) good frequency resolution Extreme Cases: (t) excellent time resolution, no frequency resolution (f) =1 excellent frequency resolution (FT), no time information 0 t (t) 0 1 F(j) dtett tj )()]([F 1 0 t tj e 1)( F t
- 7. Constant Q cf Q B f0 2f0 4f0 8f0 B 2B 4B 8B B B B B BB f0 2f0 3f0 4f0 5f0 6f0 STFTCWT
- 8. Time-frequency representations • Non-parametric methods • From the time domain to the frequency domain • Short-Time Fourier Transform - STFT • S-Transform - ST • Continuous Wavelet Transform – CWT • Synchrosqueezing transform – SST • Parametric methods • Time-series modeling (linear prediction filters) • Short-Time Autoregressive method - STAR • Time-varying Autoregressive method - KS (Kalman Smoother)
- 9. The Synchrosqueezing Transform (SST) - CWT (Daubechies, 1992) *1 ( , ) ( ) ( )sW a b s t a d a t t b Ingrid Daubechies Instantaneous frequency is the time derivative of the instantaneous phase 𝜔 𝑡 = 𝑑𝜃(𝑡) 𝑑𝑡 (Taner et al., 1979) - SST (Daubechies, 2011) the instantaneous frequency 𝜔𝑠(𝑎, 𝑏) can be computed as the derivative of the wavelet transform at any point (𝑎, 𝑏) . ( , ) ( , ) ( , ) s s s W a bj a b W a b b Last step: map the information from the time-scale plane to the time-frequency plane. (𝑏, 𝑎) → (𝑏, 𝜔𝑠(𝑎, 𝑏)), this operation is called “synchrosqueezing”
- 10. SST – Steps Synchrosqueezing depends on the continuous wavelet transform and reassignment Seismic signal 𝑠(𝑡) Mother wavelet 𝜓(𝑡) 𝑓, Δ𝑓 CWT 𝑊𝑠(𝑎, 𝑏) IF 𝑤𝑠 𝑎, 𝑏 Reassignment step: Compute Synchrosqueezed function 𝑇𝑠 𝑓, 𝑏 Extract dominant curves from 𝑇𝑠 𝑓, 𝑏 Time-Frequency Representation Reconstruct signal as a sum of modes Reassignment procedure: Placing the original wavelet coefficient 𝑊𝑠(𝑎, 𝑏) to the new location 𝑊𝑠(𝑤𝑠 𝑎, 𝑏 , 𝑏) 𝑇𝑠 𝑓, 𝑏 Auger, F., & Flandrin, P. (1995). Dorney, T., et al (2000).
- 12. Synthetic Example 1 Noiseless synthetic signal 𝑠 𝑡 as the sum of the following components: 𝑠1 𝑡 = 0.5 cos 10𝜋𝑡 , 𝑡 = 0: 6 𝑠 𝑠2 𝑡 = 0.8 cos 30𝜋𝑡 , 𝑡 = 0: 6 𝑠 𝑠3(𝑡) = 0.7 cos 20𝜋𝑡 + sin(𝜋𝑡) , 𝑡 = 6: 10.2 𝑠 𝑠4(𝑡) = 0.4 cos 66𝜋𝑡 + sin(4𝜋𝑡) , 𝑡 = 4: 7.8 𝑠 𝑓𝑠1 (𝑡) = 5 𝑓𝑠2 (𝑡) = 15 𝑓𝑠3 𝑡 = 10 + cos 𝜋𝑡 /2 𝑓𝑠4 𝑡 = 33 + 2 ∗ cos 4𝜋𝑡 𝑓 𝑡 = 𝑑𝜃(𝑡) 2𝜋 𝑑𝑡 0 2 4 6 8 10 Signal Component 4 Component 3 Component 2 Component 1 Synthetic 1 Time [s]
- 13. Synthetic Example 1: STFT vs SST Noiseless synthetic signal 𝑠 𝑡 as the sum of the following components: 𝑓𝑠1 (𝑡) = 5 𝑓𝑠2 (𝑡) = 15 𝑓𝑠3 𝑡 = 10 + cos 𝜋𝑡 /2 𝑓𝑠4 𝑡 = 33 + 2 ∗ cos 4𝜋𝑡 𝑓 𝑡 = 𝑑𝜃(𝑡) 2𝜋 𝑑𝑡 𝑠1 𝑡 = 0.5 cos 10𝜋𝑡 , 𝑡 = 0: 6 𝑠 𝑠2 𝑡 = 0.8 cos 30𝜋𝑡 , 𝑡 = 0: 6 𝑠 𝑠3(𝑡) = 0.7 cos 20𝜋𝑡 + sin(𝜋𝑡) , 𝑡 = 6: 10.2 𝑠 𝑠4(𝑡) = 0.4 cos 66𝜋𝑡 + sin(4𝜋𝑡) , 𝑡 = 4: 7.8 𝑠
- 14. Synthetic Example 2 - SST The challenging synthetic signal: - 20 Hz cosine wave, superposed 100 Hz Morlet atom at 0.3 s - two 30 Hz zero phase Ricker wavelets at 1.07 s and 1.1 s, - three different frequency components between 1.3 s and 1.7 s of respectively 7, 30 and 40 Hz. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time (s) Amplitude Synthetic Example 2
- 15. STFT SST The challenging synthetic signal: - 20 Hz cosine wave, superposed 100 Hz Morlet atom at 0.3 s - two 30 Hz zero phase Ricker wavelets at 1.07 s and 1.1 s, - three different frequency components between 1.3 s and 1.7 s of respectively 7, 30 and 40 Hz. Synthetic Example 2 – STFT vs SST
- 16. Signal Reconstruction - SST 1 - Difference between the original signal and the sum of the modes 2- Mean Square Error (MSE) 𝑀𝑆𝐸 = 1 𝑁 |𝑠 𝑡 − 𝑠(𝑡)|2 𝑁−1 𝑛=0
- 17. The challenging synthetic signal: - 20 Hz cosine wave, superposed 100 Hz Morlet atom at 0.3 s - two 30 Hz zero phase Ricker wavelets at 1.07 s and 1.1 s, - three different frequency components between 1.3 s and 1.7 s of respectively 7, 30 and 40 Hz. MSE = 0.0021 Signal Reconstruction - SSTIndividualComponents 0.5 1 1.5 2 -1 0 1 2 Time (s) Amplitude Orginal(RED) and SST estimated (BLUE) 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 Reconstruction error with SST Time (s) Amplitude -1 0 1 Mode1 SST Modes -1 0 1 Mode2 -1 0 1 Mode3 -1 0 1 Mode4 -1 0 1 Mode5 -1 0 1 Mode6 0 0.5 1 1.5 2 -1 0 1 Mode7 Time (s)
- 18. Real Example: TFR. 5 minutes of data 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -1 0 1 2 x 10 -3 Time [min] Amplitude[Volts]
- 19. Real Example. Rolla Exp. Stage A2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 x 10 4 Amplitude Time (s) s(t)
- 20. Real Example. Rolla Exp. Stage A2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 x 10 4 Amplitude Time (s) s(t)
- 21. Real Example. Rolla Exp. Stage A2 0.2 0.4 0.6 0.8 1 -1 0 1 x 10 4 Original trace Amplitude 0.2 0.4 0.6 0.8 1 -5000 0 5000 Mode 1 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -6000 -4000 -2000 0 2000 4000 Mode 2 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -1000 0 1000 Mode 3 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -500 0 500 Mode 4 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -1 0 1 Mode 5Amplitude Time (s)
- 22. More applications of SST: seismic reflection data Seismic dataset from a sedimentary basin in Canada Original data. Inline = 110 Time(s) CMP 50 100 150 200 0.2 0.4 0.6 0.8 1 1.2 1.4 Erosional surface Channels Strong reflector
- 23. CMP 81- Located at the first channel More applications of SST: seismic reflection data 0.2 0.4 0.6 0.8 1 1.2 1.4 -1.5 -1 -0.5 0 0.5 1 1.5 x 10 4 Time (s) Amplitude CMP 81
- 24. CMP 81- Located at the first chanel More applications of SST: seismic reflection data
- 25. Time slice at 420 ms SST - 20 Hz Inline (km) Crossline(km) 1 2 3 4 5 1 2 3 4 5 SST - 40 Hz Inline (km) Crossline(km) 1 2 3 4 5 1 2 3 4 5 SST - 60 Hz Inline (km) Crossline(km) 1 2 3 4 5 1 2 3 4 5 Frequency decomposition More applications of SST: seismic reflection data Channel Fault
- 26. Original data. Inline = 110 Time(s) CMP 50 100 150 200 0.2 0.4 0.6 0.8 1 1.2 1.4 Erosional surface Channels Strong reflector Frequency Frequency More applications of SST: seismic reflection data C80 Spectral Energy
- 27. Conclusions • SST provides good TFR. – Recommended for post processing and high precision evaluations. – Attractive for high-resolution time-frequency analysis of microseismic signals. • SST allows signal reconstruction – SST can extract individual components. • Applications of Signal Analysis and Reconstruction – Instrument Noise Reduction, – Signal Enhancement – Pattern Recognition
- 28. Acknowledgment Thanks to the sponsors of the For their financial support.