A proposal for regularized inversion for an ill-conditioned deconvolution operator.
Abstract
From the inverse problem theory aspect, deconvolution can be understood as the linear inversion of an ill-posed and ill-conditioned problem. The ill-conditioned property of the deconvolution operator make the solution of inverse problem sensitive to errors in the data. Tikhonov regularization is the most commonly used method for stability and uniqueness of the solution. However, results from Tikhonov method do not provide sufficient quality when the noise in the data is strong. This work uses the conjugate gradient method applied to the Tikhonov deconvolution scheme, including a regularization parameter calculated iteratively and based on the improvement criterion of Morozov discrepancy applied on the objective function. Using seismic synthetic data and real stacked seismic data, we carried out a deconvolution process with regularization and without regularization based on a conjugated gradient algorithm. A comparison of results is also presented. Applying regularized deconvolution on synthetic data shows improved stability on the solution. Additionally, real post-stack seismic data shows a direct application for increasing the vertical resolution even with noisy data.
References
https://doi.org/10.1088/0266-5611/25/1/015015
Chen, Z., Wang, Y. & Chen, X. (2012). Gabor deconvolution using regularized smoothing. SEG Technical Program Expanded Abstracts, 2012: 1-5.
https://doi.org/10.1190/segam2012-0665.1
Engl, H., Hanke, M. & Neubauer, A. (1996). Regularization of inverse problems. Dordrecht: Kluwer Academic Publishers.
https://doi.org/10.1007/978-94-009-1740-8
Fomel, S. (2007). Shaping regularization in geophysical-estimation problems. Geophysics, 72(2), R29-R36.
https://doi.org/10.1190/1.2433716
Hadamard, J. (1923). Lectures on Cauchy's problem in linear differential equations. New Haven: Yale University Press.
Hansen, C. (2002). Deconvolution and regularization with Toeplitz matrices. Numerical Algorithms, 29(4), 323-378.
https://doi.org/10.1023/A:1015222829062
Hansen, C. (2010). Discrete inverse problems: Insight and Algorithms. Philadelphia: SIAM, Society for Industrial and Applied Mathematics.
https://doi.org/10.1137/1.9780898718836
Hestenes, M. R. & Stiefel, E. L. (1952). Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards, 49(6), 409-436.
https://doi.org/10.6028/jres.049.044
Karsli, H., Guney, R. & Senkaya, M. (2012). High resolution deconvolution by combining Fx filtering and cauchy regularization. 5th Saint Petersburg International Conference & Exhibition, EAGE.
https://doi.org/10.3997/2214-4609.20143702
Leinbach, J. (1995). Wiener spiking deconvolution and minimum-phase wavelets: A tutorial. The Leading Edge, 14(3), 189-192.
https://doi.org/10.1190/1.1437110
Montenegro, A. F. (2010). Regularización de problemas inversos e imágenes borrosas. Tesis de Maestría, Departamento de Geofísica, Universidad Nacional de Bogotá, Bogotá, Colombia, 87pp.
Morozov, V. A. (1984). Methods for solving incorrectly posed problems. New York: Springer.
https://doi.org/10.1007/978-1-4612-5280-1
Sen, M. & Roy, I. (2003). Computation of differential seismograms and iteration adaptive regularization in prestack waveform inversion. Geophysics, 68(6), 2026-2039.
https://doi.org/10.1190/1.1635056
Sen, M. K. (2006). Seismic inversion. Austin: SPE.
Shewchuk, R. J. (1994). An introduction to the Conjugate Gradient method without the agonizing pain. Report, School of Computer Science, Carnegie Mellon University, Pittsburgh, USA.
Tikhonov, A. N. (1963). Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl., 4: 1035-1038.
Tikhonov, A. N. & Arsenin, V. Y. (1977). Solutions of ill-posed problems. Michigan: W. H. Winston.
Van der Baan, M. & Pham, D. T. (2008). Robust wavelet estimation and blind deconvolution of noisy surface seismic. Geophysics, 73(5), 37-46.
https://doi.org/10.1190/1.2965028
Wang, J., Wang, X. & Perz, M. (2006). Structure preserving regularization for sparse deconvolution. SEG Annual Meeting, New Orleans, Louisiana. Conference Paper 2006-2072.
https://doi.org/10.1190/1.2369944
Yilmaz, Ö. (1987). Seismic data processing in geophysics. Tulsa: Society of Exploration Geophysicists.
Zala, C. A. (1992). High-resolution inversion of ultrasonic traces. IEEE Transactions on Ultrasonics, 39(4), 438-463.
https://doi.org/10.1109/58.148535