L1 norm inversion method for deconvolution in attenuating media

Abstract

A B S T R A C TIn order to perform a good pulse compression, the conventional spike deconvolution method requires that the wavelet is stationary. However, this requirement is never reached since the seismic wave always suffers high-frequency attenuation and dispersion as it propagates in real materials. Due to this issue, the data need to pass through some kind of inverse-Q filter. Most methods attempt to correct the attenuation effect by applying greater gains for high-frequency components of the signal. The problem with this procedure is that it generally boosts high-frequency noise. In order to deal with this problem, we present a new inversion method designed to estimate the reflectivity function in attenuating media. The key feature of the proposed method is the use of the least absolute error (L1 norm) to define both the data and model error in the objective functional. The L1 norm is more immune to noise when compared to the usual L2 one, especially when the data are contaminated by discrepant sample values. It also favours sparse reflectivity when used to define the model error in regularization of the inverse problem and also increases the resolution, since an efficient pulse compression is attained. Tests on synthetic and real data demonstrate the efficacy of the method in raising the resolution of the seismic signal without boosting its noise component.

Iteratively re‐weighted and refined least squares algorithm for robust inversion of geophysical data

“…The trace is generated by convolving a Ricker wavelet with the impulse response of an earth model having a sparse reflectivity model and constant quality factor model (). Our purpose for presenting non‐stationary deconvolution is to test the performance of the algorithm on a severely ill‐conditioned problem (Oliveira and Lupinacci ; Gholami ). Therefore, the inversion is given the correct Q value and the wavelet, and hence, the forward operator used in the inversion is the same as that used for generating the trace.…”

Some remarks on Q‐compensated sparse deconvolution without knowing the quality factor Q

“…They also need to define a stabilization parameter of the inverse Q filtering operator. Oliveira and Lupinacci () proposed a ℓ 1 ‐norm inversion method for deconvolution in attenuating media. A key feature of their method is using the ℓ 1 norm to define both the data and model error in the objective function.…”

“…). The wavelet is additionally required for the methods described by Zhang and Ulrych (), van der Baan (), Oliveira and Lupinacci (), Chai et al . (, , ), and Wang et al .…”

“…Wang 2006;van der Baan 2012;Braga and Moraes 2013;Wang and Chen 2014;. The wavelet is additionally required for the methods described by Zhang and Ulrych (2007), van der Baan (2012), Oliveira and Lupinacci (2013), Chai et al (2014Chai et al ( , 2016Chai et al ( , 2018, and Wang et al (2018a).…”

Evaluation of the Sensitivity of Seismic Inversion Algorithms to Different Statistically Estimated Wavelets

“…The work of [15] seeks a solution to this problem through a strategy based on a simulated annealing algorithm. More recent studies have evolved to more efficient methods at the cost of more complexity as [16], [17]and [18].Sensitivity of the inversion algorithms was analyzed for wavelets estimated using the RESEARCH ARTICLE OPEN ACCESS [12]and a combination of smoothing the seismogram's amplitude spectrum and setting a constant linear model to the phase spectrum, similar to the methodology used in [19]. To reduce uncertainty in the phase estimate, after estimation a correction in the phase spectrum based on [20] was applied.…”