Instantaneous frequency computation: theory and practice

Transcription

1 Instantaneous requency computation: theory and practice Matthew J. Yedlin 1, Gary F. Margrave 2, Yochai Ben Horin 3 1 Department o Electrical and Computer Engineering, UBC 2 Geoscience, University o Calgary 3 Soreq, Yavneh, Israel

2 Why study the instantaneous requency? Since the paper o Taner et al (1979) this attribute has been used to interpret seismic data and relate the interpretation o meaningul geological signatures. Phase o the rectiied trace parameter as an aid to seismic interpretation ( B. Stebens, R. Parsons, D. Terral, R.T. Baumel and M. Yedlin) 6 patents granted to Conoco Inc. in: Australia, Belgium-France-Netherlands, Canada, West Germany, Great Britain, U.S.A [Feb. 1988].

3 Outline o Presentation: 1. Classical Instantaneous Frequency 2. Fomel s Improvement 3. Gabor and Cohen and Stockwell 4. Data examples 5. Conclusions

4 1. Classical Instantaneous Frequency Start with iω t e = cos( ωt) + isin ( ωt) Interpretation? Inspiration or the analytic signal: A() t = () t + ig() t where gt ( ) is the Hilbert transorm o ( t) : 1 () = ( ). ( ) gt τ dτ π t τ Interpretation?

5 Now deine (by analogy): 2 2 Envelope: Env(t)= ( t) + g( t) gt () Phase: Φ ( t) = tan 1 and () t Frequency: Ω ( t) = = dφ() t dt dg() t d () t () t g() t dt dt 2 2 () t + g() t

6 dg() t d () t () t g() t nt () Ω () t = = dt dt dt () () t 2 + g() t 2 has serious problems! These are: 1. Numerator problems non-physical results 2. Denominator problems can lead to instability Need to ix these!

7 2. Fomel s Improvement (27) Ω() t n() t Write ( t) = = in discrete inst 2π 2 πdt ( ) matrix-vector orm: Instability is obvious! =D 1 Stabilize this using Tikhonov regularization: =D 1 becomes ( 2 ) + ε 1 = D R n, where R is a n regularization operator n

8 3. Gabor, Cohen and Stockwell Dennis Gabor proposed (1946) proposed the expansion o a wave in terms o Gaussian wave packets. The mathematics or the continuous Gabor transorm emerged quickly. The discrete Gabor transorm emerged in the 198s due to Bastiaans. =

9 The Gabor Idea Dennis Gabor proposed (1946) proposed the expansion o a wave in terms o Gaussian wave packets. This is eectively a al Fourier transorm. =

10 The Gabor Idea A seismic signal Multiply A shited Gaussian A Gaussian slice or wave packet.

11 The Gabor Idea A seismic signal A suite o Gaussian slices Remarkably, the suite o Gaussian slices can be designed such that they sum to recreate the original signal with high idelity.

12 The Gabor Idea The Gabor transorm Window center time Fourier transorm Window center time time Suite o Gabor slices requency Gabor spectrum or Gabor transorm

13 Window center time The Gabor Idea The inverse Gabor transorm done two ways requency time Inverse Fourier transorm Window center time Sum Inverse Fourier transorm (Window) & Sum

14 The Continuous Gabor Transorm 2 2 ( τ t) /2σ iπ 2 τ SGabor (, t ) = g( τ) e e dτ where σ is ixed. report typo eq.16 no Following Cohen s theorem (1995) eqns ( ) [irst moment or power centroid]: NYQ 2 SGabor (, t ) d () = inst () σ NYQ 2 SGabor (, t ) d t t as

15 But the variance o this mean tends to ininity! There is clearly a problem with the concept o instantaneous requency! We can compute the al requency by using the Stockwell transorm instead o the Gabor transorm in the centroid calculation. The Stockwell transorm is given by 1 2π 2 2 ( τ t) /2 iπ2 τ S (, t ) = Stockwell g( τ) e e dτ

16 4. Data Examples We will now apply the oregoing theory to our data examples: 1. Chirp: 1 1 Hz; 2. Two sine waves: 4 Hz + 6 Hz; 3. Nonstationary seismic trace; 4. Quarry blast.

17 Figure 1. Chirp Signal 1 (a) Instantaneous Frequency 2 1 (b) (c) Fomel Gabor 5 omel gabor Seconds

18 Figure 2. Two sine waves 4 and 6 Hz 2 (a) Instantaneous Frequency 1 (b) x (c) Fomel Gabor 4 2 omel gabor Seconds

19 Figure 3. Non-stationary seismic trace.2 (a) Instantaneous Frequency 4 2 (b) (c) Fomel Gabor 4 2 omel gabor Seconds

20 Figure 4. Quarry Blast (a) x Instantaneous Frequency 5 (b) Fomel Gabor Stockwell Seconds (c)

21 Zoom o Blast Analysis (c) Fomel Gabor Stockwell

22 5. Conclusions 1. There is instability in the instantaneous requency it can go negative or have a value greater than the Nyquist requency; 2. Cohen s theorem provides an intuitive connection between instantaneous requency and the Gabor spectrum; 3. We need to ind an objective means o choosing the Gaussian window; 4. We need to ind an objective means or choosing the optimum amount o regularization in the Fomel method.