A nonstationary sparse spike deconvolution with anelastic attenuation

Abstract

Seismic wavelet estimation and deconvolution are essential for high-resolution seismic processing. Because of the influence of absorption and scattering, the frequency and phase of the seismic wavelet change with time during wave propagation, leading to a time-varying seismic wavelet. To obtain reflectivity coefficients with more accurate relative amplitudes, we should compute a nonstationary deconvolution of this seismogram, which might be difficult to solve. We have extended sparse spike deconvolution via Toeplitz-sparse matrix factorization to a nonstationary sparse spike deconvolution approach with anelastic attenuation. We do this by separating our model into subproblems in each of which the wavelet estimation problem is solved by the classic sparse optimization algorithms. We find numerical examples that illustrate the parameter setting, noisy seismogram, and the estimation error of the [Formula: see text] value to validate the effectiveness of our extended approach. More importantly, taking advantage of the high accuracy of the estimated [Formula: see text] value, we obtain better performance than with the stationary Toeplitz-sparse spike deconvolution approach in real seismic data.

Warped mapping–based blind deconvolution for resolution improvement

“…We extended the poststack nonstationary blind deconvolution (TSFM‐Q) (Sui and Ma., 2019) to the prestack CMP gather, and the objective function of the prestack Toeplitz‐sparse matrix factorization (PTSMF‐Q) was formulated as $$\begin{equation}J = \mathop {\min }\limits_{{{\bf W,r}}} \frac{1}{2}\left\| {{{\bf h}} - {{\bf WBr}}} \right\|_2^2 + {\mathop{\rm Re}\nolimits} {g_{\rm{r}}}\left( {{\bf r}} \right) + {\mathop{\rm Re}\nolimits} {g_{\rm{w}}}\left( {{\bf W}} \right).\end{equation}$$where $${\mathop{\rm Re}\nolimits} {g_{\rm{r}}}( {{\bf r}} ) = \lambda {\| {{\bf r}} \|_1}$$ represents the conventional $${L_{\rm{1}}}{\rm{ - norm}}$$ regularization for improved sparseness and $${\mathop{\rm Re}\nolimits} {g_{\rm{w}}}( {{\bf W}} )$$ represents a fused‐lasso minimum problem (Tibshirani et al ., 2005; Wang et al ., 2016; Sui and Ma., 2019) for improved balance between the sparseness and smoothness of wavelet. However, it was not easy to regularize $${{\bf W}}$$ directly.…”

“…We extended the poststack nonstationary blind deconvolution (TSFM-Q) (Sui and Ma., 2019) to the prestack CMP gather, and the objective function of the prestack Toeplitz-sparse matrix factorization (PTSMF-Q) was formulated as…”

“…where Reg r (r) = λ r 1 represents the conventional L 1 −norm regularization for improved sparseness and Reg w (W) represents a fused-lasso minimum problem (Tibshirani et al, 2005;Wang et al, 2016;Sui and Ma., 2019) for improved balance between the sparseness and smoothness of wavelet. However, it was not easy to regularize W directly.…”

“…A combination of the blind and nonstationary deconvolutions, nonstationary blind deconvolution stands as an effective routine for eliminating the influence of attenuation and the removal of the blurring effect caused by the seismic wavelet (Gray, 1979;Kaaresen and Taxt, 1998;Kazemi and Sacchi, 2014;Kazemi et al, 2016aKazemi et al, , 2016b. Sui and Ma (2019) presented a nonstationary sparse-spike deconvolution approach (TSMF-Q) with anelastic attenuation according to TSMF, and Lari and Gholami (2019) formulated nonstationary multichannel blind deconvolution by extending the convolution model of a stationary trace to nonstationary regimes using a patching approach.…”

“…However, the resolution of original field data does not usually meet the above requirement due to the anelasticity of the subsurface medium and the influence of the seismic wavelet. In recent years, to improve the resolution of seismic data, geophysicists have increasingly shown interests in Q‐compensated processing schemes, such as Q‐compensated reverse time migration (QRTM) (Zhu et al ., 2014; Zhu and Harris, 2015; Sun and Zhu, 2015; Sun et al ., 2016), inverse Q filtering (Wang, 2002, 2008; Braga and Moraes, 2013; Xue et al ., 2019) and nonstationary deconvolution (Margrave et al ., 2011; Chen et al ., 2013; Gholami, 2016, 2017; Sui and Ma, 2019). Even though QRTM shows good performance, it can be computationally more expensive than inverse Q filtering and nonstationary deconvolution.…”