The subterranean cocktail party: Identifying your seismic source among multiple random ones with time delays

Transcription

1 The subterranean cocktail party: Identifying your seismic source among multiple random ones with time delays Pawan Bharadwaj, Laurent Demanet, Aimé Fournier Department of Earth, Atmospheric and Planetary Sciences Department of Mathematics MIT Earth Resources Laboratory May 3, 27

2 Using the drill bit as a source (sometimes called inverse vertical seismic profiling VSP), the surface record comprises a sum of responses that need to be separated. (Aminzadeh and Dasgupta, 23) They may be separated by recording the vibrations of the drill string that impart energy to the bit and correlating it with the surface record. Downhole sources Slide 2

3 Downhole sources :.~:~ ::: TopSub Besides inverse- VSP, another application is when there are controlled active sources in the drill string, sometimes called seismic while drilling (SWD e.g., Joyce et al., 2). These could be broadband (pulses) or narrowband (vibrators). Desirable: use drill- bit and active sources simultaneously. Source - Receiver Array Offset =.66 ft. Measure Point Offset = 7.8 ft.... i~il i!! ~::~ ~: :-.~... ::::::: :::!,il!i i Omni-Directional Source Section Isolator Omni-Directional Receiver Array Section (24) 32.5' ft BottomSub ~v Slide 3

4 Independent Component Analysis D (w) B A = H. D nr (w) B(w), () S(w) Our ICA () comprises: measured blended signals D r (ω), r =,, n r ; 2 sources B(ω) and S(ω) to be determined; and a mixing matrix H = (H,H 2 ) also TBD. H B(ω) and H 2 S(ω) compose contributions of drill- bit and active sources to all received signals. Slide 4

5 Physical model: Born approx n (2) for wave eq. D r (w) = G(x b,x r ;w)b(w) Z + G(x b,x;w)m(x)g(x,x r ;w)b(w)dx x +G(x s,x r ;w)s(w) Z + G(x s,x;w)m(x)g(x,x r ;w)s(w)dx, (2) x Drill- bit and active source are at x b and x s, resp. G(,x;ω) denotes the Green function. m(x) is the subsurface reflectivity inverse problem if B(ω) and S(ω) would be known. Slide 5

6 D (w) B A D nr (w) {z } D(w) ICA in the frequency domain = A b, (w) A b,nr (w) {z } A(w) A s, (w) C B(w). A. (3) S(w) A s,nr (w) {z } S(w) The simplest ICA finds H so that B(ω) and S(ω) are statistically independent. H being ω- independent is unrealistic; Born approx n convolutive model H A(ω) (3). Deblending : Q b (ω) = A b (ω)b(ω), Q s (ω) = A s (ω)s(ω). Deconvolution is finding B(ω) and S(ω). Slide 6

7 Extended models A j,r (w) = G(x j,x r ;w) Z + G(x j,x;w)m(x)g(x,x r ;w)dx, (4) x Here j can be b for the drill- bit or s for the active source. (4) enables the Marmousi test below. It s formally straightforward (but complicated) to extend this to handle multiple scattering. Slide 7

8 Binning in frequency D r (ω), ω Ω, windowed Fourier transform of duration T = 2π/ ω applied to field data. Choose Ω a Ω so that ω Ω a Ω A(ω) H for all ω Ω a. Large T is required. Conventional ICA unmixing matrix W so that the band- passed source vector is recovered from the band- passed signal: S a = LPWD a, where o the diagonal scaling matrix L and o the permutation matrix P must be determined somehow (Low et al. 24, Makino et al. 25, Matsuoka et al. 22). Slide 8

9 Source models B(w)= X i X i sinc(t [w 2p T i]), X i N(,s 2 ) (6) We model the drill- bit signal as a Gaussian random process (6). Active source is similar, except its independent coefficients Y j must not be Gaussian. o For testing, we used uniformly random 2 < Y j < 2. We could extend this to colored noise σ σ i, but simply tested with σ =. Slide 9

10 The figure shows relative root- mean- square value (ordinate) of the s (cyan) and b (red) rows of S LPWD, vs number of samples per Ω a Simple synthetic test 2 (abscissa) S B Slide

11 RMS (ordinate) of S LPWD as before, vs Ω a (abscissa). Ω a must be large enough to enable sampling but small enough to capture A(ω) variability. Marmousi synthetic test 2 Q s,r Q b,r Slide

12 Active (.75 km deep) and drill (.245 km) sources ( west ) and receivers ( km east ) located in the Marmousi model. (4) A for testing, using Ricker wavelet peaked at 2 Hz. Data sampled in Ω = [8.7,2.3] Hz for Marmousi- block synthetic test T =.2 4 s Slide 2

13 For r =, actual (solid) and estimated (dashed) deblended signals for active (cyan) and drill- bit (red) sources. Every th i ω value is plotted. Marmousi- block test results Real Part Real Part Slide 3 b) A s, (w)s(w): Actual Vs. Estimated Actual ICA Est. c) A b, (w)b(w): Actual Vs. Estimated Actual ICA Est Frequency (Hz)

14 Marmousi- block results Actual (abscissa) and estimated (ordinate) deblended signals for active (cyan) and drill- bit (red) sources. Every 4 th i ω value is plotted. Samples per Ω a shown. ICA Estimate ICA Estimate ICA Estimate.6Hz (248 Samples).4Hz (52 Samples).Hz (28 Samples) Actual.6Hz (248 Samples) Figure 4: For the convolutive mixture in the Marmousi model: scatter Slide 4 plots between the deblended records Q and the actual drill bit (red) and active source (cyan) records for different frequency.4hz (52 Samples).Hz (28 Samples) Actual

15 SUMMARY We considered the deblending problem for seismic data from simultaneous random sources at different locations. As an example, SWD experiments use active drill- string sources and receivers to look around and ahead of the borehole, but these receivers also record noise from the operation of the drill bit. A conventional method for deblending is independent component analysis (ICA), which assumes a cocktail- party mixing model where each receiver records a linear combination of source signals, assumed to be statistically independent. In this talk, we sketch an extension of ICA applicability to seismic shot records with temporally convolutional mixing models with unknown wave kinematics. For more information: see authors abstract Deblending random seismic sources via independent component analysis accepted for the 27 SEG Annual Meeting. Thanks to Statoil ASA for support, and posing challenging problems! Slide 5