REFERENCES

Abma, R., 1995, Least-squares separation of signal and noise with multidimensional filters: Ph.D. thesis, Stanford University.

Bube, K. P., and Langan, R. T., 1997, Hybrid l1/l2 minimization with applications to tomography: Geophysics, 62, no. 04, 1183-1195.

Chavent, G., and Plessix, R., 1999, An optimal true-amplitude least-squares prestack depth-migration operator: Geophysics, 64, no. 2, 508-515.

Claerbout, J. F., and Fomel, S., 1999, Geophysical Estimation with Example: Class notes, http://sepwww.stanford.edu/sep/prof/index.html.

Claerbout, J. F., and Muir, F., 1973, Robust modeling with erratic data: Geophysics, 38, 820-844.

Claerbout, J. F., 1992, Earth Soundings Analysis, Processing versus Inversion: Blackwell Scientific Publication.

Claerbout, J., 1998, Multidimensional recursive filters via a helix: Geophysics, 63, no. 05, 1532-1541.

Clapp, R. G., and Brown, M., 2000, (t-x) domain, pattern-based multiple separation: SEP-103, 201-210.

Crawley, S., 1999, Interpolation with smoothly nonstationary prediction-error filters: SEP-100, 181-196.

Forgues, E., and Lambare, G., 1997, Resolution of multi-parameter ray+borne inversion: 61st Mtg. Eur. Assoc. Expl Geophys, Extended Abstracts, Session:P115.

Guitton, A., and Symes, W. W., 1999, Robust and stable velocity analysis using the Huber function: 69th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1166-1169.

Guitton, A., 2000a, Huber solver versus IRLS algorithm for quasi L1 inversion: SEP-103, 255-271.

Guitton, A., 2000b, Prestack multiple attenuation using the hyperbolic Radon transform: SEP-103, 181-201.

Harlan, W. S., 1986, Signal-noise separation and seismic inversion: Ph.D. thesis, Stanford University.

Huber, P. J., 1973, Robust regression: Asymptotics, conjectures, and Monte Carlo: Ann. Statist., 1, 799-821.

Lailly, P., 1983, The seismic inverse problem as a sequence of before stack migrations: Soc. Indust. Appl. Math., Conference on inverse scattering, 206-220.

Nemeth, T., 1996, Imaging and filtering by Least-Squares migration: Ph.D. thesis, The university of Utah.

Nichols, D., 1994, Velocity-stack inversion using ${\bf L_p}$ norms: SEP-82, 1-16.

Paige, C. C., and Saunders, M. A., 1982, LSQR: an algorithm for sparse linear equations and sparse least squares: ACM Transactions on Mathematical Software, 61, 43-71.

Spitz, S., 1999, Pattern recognition, spatial predictability, and subtraction of multiple events: The Leading Edge, 18, 55-58.

Tarantola, A., 1987, Inverse Problem Theory: Elsevier Science Publisher.

Thorson, J. R., 1984, Velocity stack and slant stack inversion methods: Ph.D. thesis, Stanford University.

 

compdatF
Figure 2 Filtering method. Input (left) and remodeled data after inversion. The maximum offset is 2.2km. Middle: fitting goal of Equation 9. Right: fitting goal of Equation 11

 

compresF
Figure 3 Filtering method. Residuals ${\bf r = \tilde{d} - d}$ after inversion.

 

compsepcresF
Figure 4 Filtering method. Spectrum of the ``simplest'' inversion residual with (left) and without PEF (right).

 

compspecS
Figure 5 Subtraction method. Spectrum of the ``simplest'' inversion residual (left) and of the subtraction scheme residual (right).

 

compmodF
Figure 6 Filtering method. Velocity domain

 

impulseF Figure 7 Filtering method. Convolution of noise with one of the inverse PEF estimated during the iterations. The coherent noise appears (dipping events).

 

compstabmod
Figure 8 Filtering method. Stability of the filtering scheme (the two right panels) as opposed to the stability of the ``simplest'' approach (the two left panels) to the number of iterations.

 

compevS
Figure 9 Subtraction method. Left: Model Space ${\bf m_s}$. Middle: Modeled noise ${\bf A_n^{-1}m_n}$. Some signal is trapped in the coherent noise due to crosstalks between H and the coherent noise PEF ${\bf A_n}$. Right: Data residual ${\bf r = \tilde{d} - d}$.

 

impulseS Figure 10 Subtraction method. Convolution of noise with the inverse PEF estimated from the data and used as the coherent noise PEF. Notice that both signal (straight lines) and noise (dipping events) are predictable, causing crosstalks with the hyperbolic Radon transform.

 

compiterS Figure 11 Convergence of the two proposed methods along with the convergence of the ``simplest'' scheme.

 

compstabmodSub
Figure 12 Subtraction method. Stability of the subtraction scheme (the two right panels) as opposed to the stability of the ``simplest'' approach (the two left panels) to the number of iterations.

 

compmod
Figure 13 Comparison study. The two proposed schemes give a better velocity panel than the ``simplest'' inversion.