High Fidelity Vibratory Seismic (HFVS): robust ...

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High Fidelity Vibratory Seismic (HFVS): Robust Inversion Using Generalized Inverse Stephen K. Chiu*, Charles W. Emmons, and Peter P. Eick, ConocoPhillips Summary One of key components of HFVS technology requires separating multiple sweeps and multi-vibrator gathers into a single source gather through a matrix inversion that involves solving a system of equations. Fast directequation solvers, such as LU decomposition, are often used to handle large volumes of 3D prestack data. However, the drawback of the fast direct-equation solvers does not reveal the uniqueness of the inverse problem and may fail to produce a satisfactory solution if the matrix is ill conditioned. Another better alternative is to use single value decomposition (SVD) to obtain a more robust leastsquares solution. But its high computational cost often limits its application to a small or moderate size of data volume. This paper shows that the use of SVD is well suited in the HFVS technology: requires minimum computational cost; provides diagnostic tools to analyze the uniqueness of the inverse problem; and produces a better source separation when the vibrator-sweep matrix is ill conditioned and a comparable result when the vibratorsweep matrix is relatively well conditioned. The synthetic and real data examples further illustrate that SVD is a preferable equation solver to be used in the HFVS system for the similar computational cost as a fast direct-equation solver. Introduction Sallas et al. (1998) described the basic components of HFVS acquisition system and theory of source separation. Krohn and Johnson (2003) integrated the inversion and signature deconvolution to design a one-step filter to separate data from multiple vibrators. They also observed that the addition of a few percent of prewhitening to the diagonal elements of the matrix helped to stabilize the inversion, but the prewhitening distorted the amplitude and phase of the wavelet. The generalized inverse obtained by SVD does not require the prewhitening in stabilizing the inversion, thus, it can minimize the amplitude and phase distortion of the inverted data. There is a rich literature on applying SVD to solve a wide range of geophysical problems (Kirlin and Done, 1999). Lines and Treitel (1984) compared the numerical accuracy between SVD and other fast linear-equation solvers and concluded that SVD method by far was the most robust and accurate in solving a least-squares problem. Although SVD is computationally expensive, it provides diagnostic tools, including matrix condition number and singular-value analysis, to analyze the stability of the inverse problem. Within the framework of HFVS technology, we compare the numerical accuracy and robustness between SVD and

SEG/Houston 2005 Annual Meeting

LU decomposition using synthetic and real data, and show how to implement SVD in solving the inverse problem and to achieve a minimum computational cost. Inversion A multi-vibrator gather is assumed to be a convolution model: vibrator sweeps that have unique phase rotations convolve with reflectivity series. The data trace for sweep i is

d i (t ) = g ij (t ) ⊗ m j (t ) , i

for

(1)

= 1 …. number of sweeps;

vibrators.

g ij

j

= 1, … number of

is the sweep i from vibrator j, and

m j , the

reflectivity model of vibrator j. In the frequency domain, equation 1 in matrix notation becomes

D( f ) = G( f ) M ( f )

.

(2)

The least-squares solution of equation 2 is

G HG M = GH D

.

(3)

The matrix inversion may be numerically unstable; an extra term

β 2I

is often added to the matrix

the inversion stable.

GHG

to make

β 2 is a constant or prewhitening, and

I is an identity matrix. This gives a stabilized leastsquares estimate (G H G + β 2 I ) M = G H D .

(4)

A fast equation solver such as LU decomposition can solve this system of equations. Another better way to solve the least-squares solution of equation 3 is to use the singular value decomposition (Aki and Richard, 1980 and Golub and Van Loan, 1996), G becomes

G =US V H ,

(5)

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HFVS: Robust Inversion Using Generalized Inverse

U is a matrix of eigenvectors that span the data space, V is a matrix of eigenvectors that span the model space, S is a diagonal singular-value matrix whose diagonal elements are called singular values, and H is a


where

conjugate transpose operator. The generalized least-squares solution of M is

M = (G H G ) −1 G H D = V S −1 U H D .

(6)

The singular-value analysis can reveal the uniqueness of the matrix to be solved. If some of the singular values are close to zero, the matrix is ill conditioned. Larger singular values usually represent signals of the data and smaller singular values correspond to noise of the data. By excluding some of the small singular values, we throw away a subset of corrupted equations to obtain a more reliable solution. The threshold of excluding small singular values is usually a few percent of the largest singular value. The solution with the threshold applied is often referred as the generalized inverse or principal component analysis.

3D data example The use of SVD is well suited in the unique HFVS acquisition geometry in terms of the inversion robustness and minimum computational cost. Typically, the recorded data per setup consist of a multi-vibrator gather repeated for multiple sweeps at the same source locations. The number of vibrators and sweeps used in the field is, in general, between 3 and 6. Thus, the maximum size of the matrix to be inverted is a 6*6 complex matrix per frequency. For each setup, we first decompose the vibratorsweep matrix into eigenvector and singular-value matrices and store them in the computer memory before processing the data. The number of receiver channels available in the field is typically in the order of thousands. As an example of our recent acquisition, we recorded up to 8000 receiver channels in a setup. Processing the 8000-recorded traces in one setup only requires a single decomposition of the vibrator-sweep matrix. Therefore, the computational cost of using the expensive SVD in HFVS is insignificant. We use this 3D data set as an example to demonstrate the efficiency and robustness of our implementation.

Numerical comparisons

Conclusions

To compare the performance of these two numerical methods in solving the least-squares problem, we generated two synthetic models, one with a relatively goodconditioned matrix, and the other is a relatively poorconditioned matrix. We only show the latter case here. Figures 1c to 1e show how various levels of added random noise affect the least-squares solution. The magnitude of random noise ranges from 0 to 200 % of the magnitude of the primary signals. In the noise free case, the result inverted by LU decomposition matches the theoretical model almost exactly (figures 1b and 1c). However, the resolution of the inversion with 5% prewhitening deteriorates as the noise level increases (figures 1d and 1e). The inversion results by SVD with 5% threshold of the largest singular value show similar degradation as the noise level increases (figures 2d and 2e), but the degradation is much less when compared to the LU decomposition (figures 1e and 2e). For the real data cases, a relatively good-designed phase encoding scheme gives virtually identical results between these two methods (figures 3a and 3b). However, for a poor-designed phase encoding scheme, LU decomposition seems to create aliased noise (circle area in figure 4a), but SVD does not have this artifact (figure 4b). In this case, the matrix condition number and singularvalue analysis also indicate that this phase encoding scheme is extremely unstable.

Synthetic and real data support that the use of SVD provides diagnostic tools to analyze the stability of the inverse problem, produces a more robust solution than LU decomposition for a poor-designed phase encoding scheme, and gives a comparable solution for a good-designed phase encoding scheme. SVD should be used as a preferable equation solver in HFVS technology, as it has the benefits of both worlds: robust inversion and minimum computational cost.


References Aki, K., and Richard, P., 1980, Quantitative seismology – theory and methods, volume 2, W.H. Freeman, San Franciso, 659-689. Golub, G.H. and Van Loan, C.F., 1996, Matrix computations, Johns Hopkins university press, Baltimore and London. Kirlin, R.L. and Done W.J., Editors, 1999, Covariance analysis for seismic signal Processing, SEG, Tulsa. Krohn, C.E. and Johnson, M.L. 2003, High fidelity vibratory seismic(HFVS) I: Enchanted data quality, 73th Annual Internat. Mtg. Soc. Expl. Geophys. Expanded Abstracts, 43-46.

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Lines, L.R. and Treitel, S., 1984, Tutorial: A review of least-squares inversion and its application to geophysical problems, Geophysical Prospecting, 32, 159-186. Sallas, J., Corrigan, D., and Allen, K.P., 1998, High fidelity vibratory source seismic method with source separation, U.S. Patent 5,721,710. Acknowledgments The authors thank ConocoPhillips for the permission to publish this work and Dan Whitmore to review this paper.

a

b

c

d

e

Figure 1. Inversion by LU decomposition: (a) synthetic data generated from 4 vibrators and 4 sweeps, (b) theoretical result, (c) inversion result of noise-free data, (d) inversion result using signal-to-noise ratio of 1:1, (e) inversion result using signal-to-noise ratio of 1:2.

a b c d e Figure 2. Inversion by SVD: (a) same synthetic data generated from 4 vibrators and 4 sweeps, (b) theoretical result, (c) inversion result of noise-free data, (d) inversion result using signal-to-noise ratio of 1:1, (e) inversion result using signal-to-noise ratio of 1:2.


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a

b

Figure 3. A portion of a 3D shot record with a good-designed phase encoding scheme, (a) inverted by LU decomposition, (b) inverted by SVD.

a

b

Figure 4. A portion of a 3D shot record with a poor-designed phase encoding scheme, (a) inverted by LU decomposition, (b) inverted by SVD.


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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. High Fidelity Vibratory Seismic (HFVS): Robust Inversion Using Generalized Inverse References Aki, K., and P. Richard, 1980, Quantitative seismology – theory and methods, volume 2: W. H. Freeman. Golub, G. H. and C. F. Van Loan, 1996, Matrix computations: Johns Hopkins University press. Kirlin, R. L. and W. J. Done, eds., 1999, Covariance analysis for seismic signal processing: SEG. Krohn, C. E. and M. L. Johnson, 2003, High fidelity vibratory seismic (HFVS) I: Enchanted data quality: 73rd Annual International Meeting, SEG, Expanded Abstracts, 43-46. Lines, L. R. and S. Treitel, 1984, Tutorial: A review of least-squares inversion and its application to geophysical problems: Geophysical Prospecting, 32, 159-186. Sallas, J., D. Corrigan, and K. P. Allen, 1998, High fidelity vibratory source seismic method with source separation: U.S. Patent 5 721 710.