Abstract
Reflectivity inversion is an important deconvolution problem in reflection seismology that helps to describe the subsurface structure. Generally, deconvolution techniques iteratively work on the seismic data for estimating reflectivity. Therefore, these techniques are computationally expensive and may be slow to converge. In this paper, a novel method for estimating reflectivity signals in seismic data using an approximate finite rate of innovation (FRI) framework, is proposed. The seismic data is modeled as a convolution between the Ricker wavelet and the FRI signal, a Dirac impulse train. Relaxing the accurate exponential reproduction limitation given by generalised Strang-Fix (GSF) conditions, we develop a suitable sampling kernel utilizing Ricker wavelet which allows us to estimate the reflectivity signal. The experimental results demonstrate that the proposed approximate FRI framework provides a better reflectivity estimation than the deconvolution technique for medium-to-high signal-to-noise ratio (SNR) regimes with nearly 18% of seismic data.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Figa_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig10_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-024-02749-4/MediaObjects/34_2024_2749_Fig11_HTML.png)
Similar content being viewed by others
Data Availability
The data that support the findings of this study are available from the corresponding author on request.
References
A. Adler, M. Araya-Polo, T. Poggio, Deep learning for seismic inverse problems: toward the acceleration of geophysical analysis workflows. IEEE Signal Process. Mag. 38(2), 89–119 (2021)
R.G. Baraniuk, Compressive sensing [lecture notes]. IEEE Signal Process. Mag. 24(4), 118–121 (2007)
T. Blu, P.L. Dragotti, M. Vetterli, P. Marziliano, L. Coulot, Sparse sampling of signal innovations. IEEE Signal Process. Mag. 25(2), 31–40 (2008)
J.A. Cadzow, Signal enhancement-a composite property map** algorithm. IEEE Trans. Acoust. Speech Signal Process. 36(1), 49–62 (1988)
L. Condat, A. Hirabayashi, Cadzow denoising upgraded: a new projection method for the recovery of Dirac pulses from noisy linear measurements. Sampling Theory Signal Image Process. 14(1), 17–47 (2015)
V. Das, A. Pollack, U. Wollner, T. Mukerji, Convolutional neural network for seismic impedance inversion CNN for seismic impedance inversion. Geophysics 84(6), R869–R880 (2019)
P.L. Dragotti, M. Vetterli, T. Blu, Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets strang-fix. IEEE Trans. Signal Process. 55(5), 1741–1757 (2007)
A. Erdozain, P.M. Crespo, A new stochastic algorithm inspired on genetic algorithms to estimate signals with finite rate of innovation from noisy samples. Signal Process. 90(1), 134–144 (2010)
A. Erdozain, P.M. Crespo, Reconstruction of aperiodic FRI signals and estimation of the rate of innovation based on the state space method. Signal Process. 91(8), 1709–1718 (2011)
M.F. Fahmy, G. Fahmy, Exponential spline perfect reconstruction, decomposition and reconstruction with applications in compression and denoising. SIViP 8, 1111–1120 (2014)
Y. Hua, T.K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process. 38(5), 814–824 (1990)
G. Huang, N. Fu, J. Zhang, L. Qiao, Sparsity-based reconstruction method for signals with finite rate of innovation. In: 2016 IEEE International Conference on Acoustics. (IEEE, Speech and Signal Processing (ICASSP), 2016), pp.4503–4507
V.C. Leung, J.J. Huang, P.L. Dragotti, Reconstruction of FRI signals using deep neural network approaches. In: ICASSP 2020–2020 IEEE International Conference on Acoustics. (IEEE, Speech and Signal Processing (ICASSP), 2020), pp.5430–5434
V.C. Leung, J.J., Huang Y.C., Eldar P.L. Dragotti, Reconstruction of FRI signals using autoencoders with fixed decoders. In: 2021 29th European Signal Processing Conference (EUSIPCO), IEEE, pp 1496–1500 (2021)
V.C. Leung, J.J. Huang, Y.C. Eldar, P.L. Dragotti, Learning-based reconstruction of FRI signals. IEEE Transactions on Signal Processing (2023)
C. Li, X. Liu, K. Yu, X. Wang, F. Zhang, Debiasing of seismic reflectivity inversion using basis pursuit de-noising algorithm. J. Appl. Geophys. 177, 104028 (2020)
I. Maravic, M. Vetterli, Sampling and reconstruction of signals with finite rate of innovation in the presence of noise. IEEE Trans. Signal Process. 53(8), 2788–2805 (2005)
I. Markovsky, Structured low-rank approximation and its applications. Automatica 44(4), 891–909 (2008)
G.S. Martin, R. Wiley, K.J. Marfurt, Marmousi2: An elastic upgrade for marmousi. Lead. Edge 25(2), 156–166 (2006)
S. Mulleti, H. Zhang, Y.C. Eldar, Learning to sample: data-driven sampling and reconstruction of FRI signals. IEEE Access 11, 71048 (2023)
M. Najjarzadeh, H. Sadjedi, Reconstruction of finite rate of innovation signals in a noisy scenario: a robust, accurate estimation algorithm. SIViP 14(8), 1707–1715 (2020)
E. Nova Scotia Department, Penobscot 3d survey, data retrieved from https://terranubis.com/datainfo/penobscot (2017)
D. Oldenburg, T. Scheuer, S. Levy, Recovery of the acoustic impedance from reflection seismograms. Geophysics 48(10), 1318–1337 (1983)
D.O. Pérez, D.R. Velis, M.D. Sacchi, High-resolution prestack seismic inversion using a hybrid fista least-squares strategy. Geophysics 78(5), R185–R195 (2013)
B. Russell, Machine learning and geophysical inversion—a numerical study. Lead. Edge 38(7), 512–519 (2019)
P.M. Shearer, Introduction to Seismology (Cambridge University Press, Cambridge, 2009)
P. Sudhakar Reddy, A. Premkumar, B. Saikiran, B.S. Raghavendra, A.V. Narasimhadhan, Finite rate of innovation signal reconstruction using residual neural networks. In: 2020 IEEE 4th Conference on Information & Communication Technology (CICT), IEEE, pp 1–6 (2020)
P. Sudhakar Reddy, B.S. Raghavendra, A.V. Narasimhadhan, Magnetic resonance image reconstruction by nullspace based finite rate of innovation framework. Proceedings of the Twelfth Indian Conference on Computer Vision, Graphics and Image Processing 1–5 (2021)
P. Sudhakar Reddy, B. Raghavendra, A. Narasimhadhan, Universal discrete finite rate of innovation scheme for sparse signal reconstruction. Circuits Syst. Signal Process. 42(4), 2346–2365 (2023)
P. Sudhakar Reddy, B.S. Raghavendra, A.V. Narasimhadhan, Sparse-prony fri signal reconstruction. Signal, Image and Video Processing 1–7 (2023)
V.Y. Tan, V.K. Goyal, Estimating signals with finite rate of innovation from noisy samples: a stochastic algorithm. IEEE Trans. Signal Process. 56(10), 5135–5146 (2008)
T. Tonellot, M. Broadhead, Sparse seismic deconvolution by method of orthogonal matching pursuit. In: 72nd EAGE Conference and Exhibition incorporating SPE EUROPEC 2010, European Association of Geoscientists & Engineers, cp–161 (2010)
J.A. Urigüen, T. Blu, P.L. Dragotti, Fri sampling with arbitrary kernels. IEEE Trans. Signal Process. 61(21), 5310–5323 (2013)
R. Versteeg, The marmousi experience: velocity model determination on a synthetic complex data set. Lead. Edge 13(9), 927–936 (1994)
M. Vetterli, P. Marziliano, T. Blu, Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50(6), 1417–1428 (2002)
J. Wang, S. Wang, S. Yuan, J. Li, H. Yin, Stochastic spectral inversion for sparse-spike reflectivity by presetting the number of non-zero spikes as a prior sparsity constraint. J. Geophys. Eng. 11(1), 015010 (2014)
Y.Q. Wang, Q. Wang, W.K. Lu, Q. Ge, X.F. Yan, Seismic impedance inversion based on cycle-consistent generative adversarial network. Pet. Sci. 19(1), 147–161 (2022)
A. Wein, L. Srinivasan, Iterml: a fast, robust algorithm for estimating signals with finite rate of innovation. IEEE Trans. Signal Process. 61(21), 5324–5336 (2013)
B. Wu, D. Meng, L. Wang, N. Liu, Y. Wang, Seismic impedance inversion using fully convolutional residual network and transfer learning. IEEE Geosci. Remote Sens. Lett. 17(12), 2140–2144 (2020)
C. Yuan, M. Su, Seismic spectral sparse reflectivity inversion based on SBL-EM: experimental analysis and application. J. Geophys. Eng. 16(6), 1124–1138 (2019)
C. Yuan, M. Su, Seismic spectral sparse reflectivity inversion based on SBL-EM: experimental analysis and application. J. Geophys. Eng. 16(6), 1124–1138 (2019)
S. Yuan, S. Wang, Spectral sparse bayesian learning reflectivity inversion. Geophys. Prospect. 61(4), 735–746 (2013)
J. Zhang, C. Huang, M.Y. Chow, X. Li, J. Tian, H. Luo, S. Yin, A data-model interactive remaining useful life prediction approach of lithium-ion batteries based on PF-BIGRU-TSAM. IEEE Transactions on Industrial Informatics (2023)
J. Zhang, J. Tian, A.M. Alcaide, J.I. Leon, S. Vazquez, L.G. Franquelo, H. Luo, S. Yin, Lifetime extension approach based on levenberg-marquardt neural network and power routing of dc-dc converters. IEEE Transactions on Power Electronics (2023)
R. Zhang, J. Castagna, Seismic sparse-layer reflectivity inversion using basis pursuit decomposition. Geophysics 76(6), R147–R158 (2011)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Authors declare that they have no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Reddy, P.S., Raghavendra, B.S. & Narasimhadhan, A.V. Approximate Finite Rate of Innovation Based Seismic Reflectivity Estimation. Circuits Syst Signal Process (2024). https://doi.org/10.1007/s00034-024-02749-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00034-024-02749-4