time-frequency representation of microseismic signals using the sst
DESCRIPTION
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.TRANSCRIPT
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
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.
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).
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
• 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
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 10
t
tje
1)( Ft
Constant Q
cfQB
f0 2f0 4f0 8f0
B 2B 4B 8B
B B B B B B
f0 2f0 3f0 4f0 5f0 6f0
STFT
C
WT
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)
The Synchrosqueezing Transform (SST)
- CWT (Daubechies, 1992)
*1
( , ) ( ) ( )sW a b s ta
da
tt
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 bja 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”
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).
Extracting curves
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]
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
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)
Am
plitu
de
Synthetic Example 2
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
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)|2𝑁−1
𝑛=0
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 - SST In
div
idua
l C
om
po
ne
nts
0.5 1 1.5 2
-1
0
1
2
Time (s)
Am
plitu
de
Orginal(RED) and SST estimated (BLUE)
0 0.5 1 1.5 2-1
-0.5
0
0.5
1Reconstruction error with SST
Time (s)
Am
plitu
de
-1
0
1
Mo
de
1
SST Modes
-1
0
1
Mo
de
2
-1
0
1
Mo
de
3
-1
0
1
Mo
de
4
-1
0
1
Mo
de
5
-1
0
1
Mo
de
6
0 0.5 1 1.5 2-1
0
1
Mo
de
7
Time (s)
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]
Am
plitu
de
[V
olts]
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 104
Am
plitu
de
Time (s)
s(t)
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 104
Am
plitu
de
Time (s)
s(t)
Real Example. Rolla Exp. Stage A2
0.2 0.4 0.6 0.8 1
-1
0
1
x 104 Original trace
Am
plitu
de
0.2 0.4 0.6 0.8 1
-5000
0
5000
Mode 1
Am
plitu
de
Time (s)
0.2 0.4 0.6 0.8 1-6000-4000-2000
020004000
Mode 2
Am
plitu
de
Time (s)
0.2 0.4 0.6 0.8 1
-1000
0
1000
Mode 3
Am
plitu
de
Time (s)
0.2 0.4 0.6 0.8 1-500
0
500
Mode 4
Am
plitu
de
Time (s)
0.2 0.4 0.6 0.8 1-1
0
1Mode 5
Am
plitu
de
Time (s)
More applications of SST: seismic reflection data
Seismic dataset from a sedimentary basin in Canada
Original data. Inline = 110
Tim
e (
s)
CMP50 100 150 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Erosionalsurface Channels
Strong reflector
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 104
Time (s)
Am
plitu
de
CMP 81
CMP 81- Located at the first chanel
More applications of SST: seismic reflection data
Time slice at 420 ms
SST - 20 Hz
Inline (km)
Cro
sslin
e (
km)
1 2 3 4 5
1
2
3
4
5
SST - 40 Hz
Inline (km)
Cro
sslin
e (
km)
1 2 3 4 5
1
2
3
4
5
SST - 60 Hz
Inline (km)
Cro
sslin
e (
km)
1 2 3 4 5
1
2
3
4
5
Frequency decomposition
More applications of SST: seismic reflection data
Channel
Fault
Original data. Inline = 110
Tim
e (
s)
CMP50 100 150 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Erosionalsurface Channels
Strong reflector
Fre
quency
Fre
quency
More applications of SST: seismic reflection data
C80 Spectral
Energy
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
Acknowledgment
Thanks to the sponsors of the
For their financial support.