Resonance frequencies could provide useful information on the deformation occurring during fracturing experiments or CO2 management, complementary to the microseismic events distribution. An accurate time-frequency representation is of crucial importance to interpret the cause of resonance frequencies during microseismic experiments. The popular methods of Short-Time Fourier Transform (STFT) and wavelet analysis have limitations in representing close frequencies and dealing with fast varying instantaneous frequencies and this is often the nature of microseismic signals. The synchrosqueezing transform (SST) is a promising tool to track these resonant frequencies and provide a detailed time-frequency representation. Here we apply the synchrosqueezing transform to microseismic signals and also show its potential to general seismic signal processing applications.

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).

11. Extracting curves

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.