Signal and traveltime parameter estimation using singular value decomposition
INCT-GP
ˆ Bjorn Ursin, Michelangelo G. Silva and Milton J. Porsani NTNU
UFBA
Norwegian University of Science and Technology (NTNU), Norwa, Centro de Pesquisa em Geof´ısica e Geologia (CPGG-UFBA), Brazil, National Institute of Science and Technology of Petroleum Geophysics (INCT-GP/CNPq/MCT), Brazil.
EAGE Meeting, Amsterdam 16-19 June, 2014
INTRODUCTION I
I
NORMALIZED COHERENCE MEASURES
Signal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters. We have used SVD of the data matrix, not eigen decomposition of a covariance matrix, to review the most commonly used coherence measures. This results in a new reduced semblance coefficient defined from the first eigenimage, assuming that the signal amplitude is the same on all data channels (as in classical semblance. In a second signal model the time signal is constant on each channel, but the amplitude changes. Then semblance coefficient is the square of the first singular value divided by the data energy. Two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as coherence measure. The normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one, and the normalized crosscorrelation of the temporal singular vector and the average time signal (the stacked trace). We define a multiple signal classification (MUSIC) measure as the inverse of one minus any of the normalized coherence measures described above. In order to reduce the numerical range we prefer to use log10MUSIC.
The classical coherence measure used in seismic velocity analysis is semblance (Taner and Koehler, 1969): ||W||2F S= ||D||2F
0≤S≤1
Signal model W = seT with eT = [1, 1, . . . , 1] the stacking vector. 1 ˆ= s De Nx
Signal estimate:
ˆ ||2 ||De||2 Nx ||s = S= 2 ||D||F Nx ||D||2F
Classical semblance:
The multiple signal classification (MUSIC) algorithm(Schmidt, 1986; Barros et al, 2012): 1 P= 1−S
Then P ≥ 1 for any S, 0 ≤ S ≤ 1.
TRAVELTIME PARAMETERS log10 P = − log10(1 − S) ≥ 0
We will use (Gulunay, 1991): We assume that the data d(t, x), function of time t and direction x, can be represented by a sum of arriving pulses, sk (t), with different arrival times T (θ k , x) which have the same form, but different values of the traveltime parameters vector θ. That is d(t, x) =
K X
SVD SUBSPACE SEMBLANCE Reduced SVD (Golub and van Loan, 1996):
sk (t − T (θ k , x)) + n(t, x)
T
D = UΣV =
k=1
uk σk vTk
Nt ≤ Nx
k=1
where n(t, x) is a noise term. The traveltime or arrival time function may have different forms. The hyperbolic traveltime approximation used in standard seismic velocity analysis (Taner and Koehler, 1969) is: #1/2 " 2 x , θ = [T (0), vNMO ], T (θ, x) = T (0)2 + 2 vNMO
SVD subspace signal: W = u1σ1vT1 Signal estimate:
For a plane wave the traveltime function (Schmidt, 1986) is: θ = [T (0), p],
where T (0) = zero-offset traveltime and p = slowness. Data window: T (θ, x) ± ((Nt − 1)/2)∆t ( j = 1, . . . , Nt Data matrix: d(tj , xn) = Djn n = 1, . . . , Nx
σ1 ˆ1 = s u1vT1 e Nx
Reduced semblance 2
ˆ 1|| Nx ||s SR = = 2 ||D||F
where vNMO = the normal moveout velocity is often approximated by vs , the stacking velocity.
T (θ, x) = T (0) + px,
Nt X
σ12|vT1 e|2 Nx ||D2||F
PNt
2 T 2 σ 2 k |vk e| =S− PNt 2 Nx k=1 σk
EIGENIMAGE SIGNAL ENERGY The signal model:
W = saT
A least-squares estimate of the signal is the first eigenimage (Golub and van Loan, 1996): W = u1σ1vT1 Semblance (Gersztenkorn and Marfurt, 1999):
σ12 σ12 = PN SE = 2 t 2 ||D||F σ k=1 k
is the sum of the signal and noise matrices: D = W + N. NORMALIZED EIGENVECTOR CROSSCORRELATION COEFFICIENTS From Barros et al(2012): The spatial covariance eigenvector gives: |vT1 e|2 1 T 2 SM = = |v1 e| 2 2 ||e|| ||v1|| Nx SR . Note: SM = SE The temporal covariance eigenvector gives: ˆ |2 |uT1 s |uT1 De|2 σ12|vT1 e|2 ST = = = ˆ ||2 ||De||2 ||De||2 ||u1||2||s Figure 1 : Data analysis window
SR . Note: ST = S
Mails:
[email protected], mykael
[email protected],
[email protected]
Websites: www.ntnu.edu, www.ufba.br, www.inct-gp.org
Signal and traveltime parameter estimation using singular value decomposition RESULTS Traces 0.8
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Figure 2 : Synthetic data for time resolution and AVO effects: Little noise (a), Medium noise (b), Much noise (c) 1
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Figure 5 : Different coherence measure in velocity analysis of marine seismic data
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Two generalized semblance functions are better than classical semblance:
Severe AVO effects hard to detect.
INCT-GP/CNPq/MCT, Petrobras, ANP, FINEP, FAPESB BRAZIL for financial support. I Statoil and the Norwegian Research Council through the ROSE project for financial support. I LANDMARK for the licenses granted to CPGG-UFBA. I
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REFERENCES
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0.60 0.58 1.2
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0.25 0.20 0.15
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0.42 0.40 0.38 1.2
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log P
0.10 1.2
1.2 0.05
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Time(s)
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Barros, T. , R. Lopes, M. Tygel, and J. T. M. Romano, 2012, Implementation aspects of eigenstructure-based velocity spectra. 74th EAGE Conference, Copenhagen, expanded abstracts. I Gersztenkorn, A., and K. J. Marfurt, 1999, Eigenstructure-based coherence conputations as an aid to 3-D structural and stratigraphic mapping: Geophysics,64, 1468-1479. I Golub, B. H. and C.F. van Loan, 1996, Matrix Computations. The Johns Hopkins University Press, Baltimore, 3rd edition. I Gulunay, N., 1991, High-resolution CVS: Generalized covariance measure: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1264-1267. I Schmidt, R., 1986, Multiple emitter location and signal parameter estimation: IEEE Trans. Antennas Propagat., 34, 276-280. I Taner, M. T., and F. Koehler, 1969, Velocity spectra: digital computer derivation and application of velocity functions: Geophysics, 34, 859-881. I
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ACKNOWLEDGEMENTS
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Log10 MUSIC of normalized temporal covariance eigenvector crosscorrelation works for high SNR.
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A reduced semblance coefficient, taking into account only the first eigenimage; I The normalized energy of the first eigenimage.
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CONCLUSIONS
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Figure 6 : Details from the velocity analysis of the previous figure have also included log10PE
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Figure 3 : Coherence measure (left panels) and log MUSIC (right panels) for time-resolution and AVO data for VNMO=3000 m/s and different times. Little-noise data (top panels), Medium-noise data (middle panels) and Much-noise data (bottom panels). The vertical dashed lines indicate the correct values of T(0)
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Value Coherence
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3 - Much-noise panel Figure 4 : AVO data: Semblance functions (top panels), and the respective log MUSIC functions (bottom panels). The crosses with dashed lines indicate the correct parameter values.
Mails:
[email protected], mykael
[email protected],
[email protected]
Websites: www.ntnu.edu, www.ufba.br, www.inct-gp.org
