Absorption-compensation method by l1-norm regularization

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. V107-V114 ◽  
Author(s):  
Shoudong Wang ◽  
Xiaohong Chen
Keyword(s):  
Integral Equation ◽  
Seismic Wave ◽  
Fredholm Integral Equation ◽  
Seismic Data ◽  
Real Data ◽  
Compensation Method ◽  
L1 Norm ◽  
Text Filtering ◽  
The Time Domain ◽  
Attenuation And Dispersion

Absorption in subsurface media greatly reduces the resolution of seismic data. Absorption not only dissipates the high-frequency components of the wave, but it also distorts the seismic wavelet. The inverse [Formula: see text]-filtering method is an effective method to correct the attenuation and dispersion of the seismic wave. We evaluated an absorption compensation method based on [Formula: see text]-norm regularization. Forward modeling in the absorption medium is described by a Fredholm integral equation in the time domain, and the absorption compensation comes down to solving the Fredholm integral equation. The solutions of the Fredholm integral equation are extremely unstable. Generally, [Formula: see text]-norm regularization can be used to seek stable solutions of the Fredholm integral equation, but it makes the solutions smooth and reduces the resolution of seismic data. We used [Formula: see text]-norm regularization to overcome instability of solving the integral equation; it makes the solutions have a higher resolution than [Formula: see text]-norm regularization. The iteratively reweighted method is used to solve the minimization problem. Tests on synthetic and real data examples showed that the absorption compensation method that we evaluated can effectively correct the attenuation and dispersion of the seismic wave.

Sparse reflectivity inversion for nonstationary seismic data

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. V93-V105 ◽  
Author(s):  
Xintao Chai ◽  
Shangxu Wang ◽  
Sanyi Yuan ◽  
Jianguo Zhao ◽  
Langqiu Sun ◽  
...  
Keyword(s):  
Seismic Data ◽  
Inversion Method ◽  
The Novel ◽  
Field Surveys ◽  
Convolution Model ◽  
Stationary Signals ◽  
Text Filtering ◽  
Nonstationary Data ◽  
Attenuation And Dispersion ◽  
Reflectivity Inversion

Conventional reflectivity inversion methods are based on a stationary convolution model and theoretically require stationary seismic traces as input (i.e., those free of attenuation and dispersion effects). Reflectivity inversion for nonstationary data, which is typical for field surveys, requires us to first compensate for the earth’s [Formula: see text]-filtering effects by inverse [Formula: see text] filtering. However, the attenuation compensation for inverse [Formula: see text] filtering is inherently unstable, and offers no perfect solution. Thus, we presented a sparse reflectivity inversion method for nonstationary seismic data. We referred to this method as nonstationary sparse reflectivity inversion (NSRI); it makes the novel contribution of avoiding intrinsic instability associated with inverse [Formula: see text] filtering by integrating the earth’s [Formula: see text]-filtering operator into the stationary convolution model. NSRI also avoids time-variant wavelets that are typically required in time-variant deconvolution. Although NSRI is initially designed for nonstationary signals, it is suitable for stationary signals (i.e., using an infinite [Formula: see text]). The equations for NSRI only use reliable frequencies within the seismic bandwidth, and the basis pursuit optimizes a cost function of mixed [Formula: see text] norms to derive a stable and sparse solution. Synthetic examples show that NSRI can directly retrieve reflectivity from nonstationary data without advance inverse [Formula: see text] filtering. NSRI is satisfactorily stable in the presence of severe noise, and a slight error in the [Formula: see text] value does not greatly disturb the sensitivity of NSRI. A field data example confirmed the effectiveness of NSRI.

Inverse Q filtering via synchrosqueezed wavelet transform

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. V121-V132 ◽  
Author(s):  
Ya-Juan Xue ◽  
Jun-Xing Cao ◽  
Xing-Jian Wang
Keyword(s):  
Wavelet Transforms ◽  
Ambient Noise ◽  
Real Data ◽  
Dispersive Media ◽  
Time Frequency ◽  
Effective Components ◽  
Text Filtering ◽  
Inverse Q Filtering ◽  
The Time Domain ◽  
Amplitude Compensation

We have developed and applied an inverse [Formula: see text]-filter formulation using synchrosqueezed wavelet transforms for the compensation of attenuating and dispersive media. A damping criterion concerning the reconstruction of the effective components for controlling noise amplification and the separation of the noise and signal in the synchrosqueezed wavelet domain is generated. The proposed method provides stable attenuation compensation without decreasing the seismic vertical and lateral resolution. The best property of the proposed method, unlike conventional inverse [Formula: see text]-filtering methods, is that it carries out amplitude compensation for the effective components located at some time samples in the time-frequency domain. The spectral reconstruction contributes to the reconstruction of the trace in the time domain and suppresses the ambient noise located at high frequencies at later times, especially suppressing the ambient noise within the main frequency band. It is not a noise-level-dependent method. We validated our approach with synthetic and real data. The comparison of the proposed method with the conventional stabilized inverse [Formula: see text]-filtering method is also carried out to illustrate the particular features of the proposed method. The examples demonstrate that our proposed method can effectively compensate for the amplitude attenuation by suppressing the ambient noise and further provide seismic images at high resolution while highlighting the effective details. Furthermore, it is a robust and easily tunable algorithm.

Modeling of attenuation and dispersion

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1167-1173 ◽  
Author(s):  
Carlos Lopo Varela ◽  
Andre L. R. Rosa ◽  
Tadeusz J. Ulrych
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Low Cost ◽  
Real Data ◽  
Dispersion Modeling ◽  
Minimum Phase ◽  
The Time Domain ◽  
Wavefield Extrapolation ◽  
Attenuation And Dispersion ◽  
Proper Solutions

At the present time, proper solutions for absorption modeling are based on wavefield extrapolation techniques which, in some instances, may be considered expensive. Two alternative, low cost, but incomplete solutions exist in the literature. The first models dispersion in the frequency domain in accordance with the Futterman dispersive relations but does not consider attenuation. The second models both attenuation and dispersion in the time domain but assumes a digital minimum‐phase formulation that results in an inadequate treatment of the dispersion. We show that this second solution can be adapted to perform attenuation and/or dispersion modeling in agreement with the Futterman attenuation‐dispersion relationships thus obviating the shortcoming mentioned above. Synthetic and real data examples are shown to illustrate the performance of the proposed algorithm.

A depth variant seismic wavelet extraction method for inversion of poststack depth-domain seismic data

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. R569-R579 ◽  
Author(s):  
Rui Zhang ◽  
Zhiwen Deng
Keyword(s):  
Seismic Data ◽  
Extraction Method ◽  
Reservoir Characterization ◽  
Real Data ◽  
Velocity Model ◽  
Seismic Inversion ◽  
Data Depth ◽  
Spectral Variation ◽  
Seismic Wavelet ◽  
The Time Domain

Prestack depth seismic imaging is increasingly being used in industry, which has also led to an increasing need for its inversion results, such as acoustic impedance (AI), for reservoir characterization. Conventional seismic inversion methods for reservoir characterization are usually implemented in the time domain. A depth-time conversion would be required before inversion of depth-domain seismic data, which would depend on an accurate velocity model and a fine time-depth conversion algorithm. Thus, it could be beneficial that we can directly invert the depth migrated seismic data. Depth-domain seismic data could indicate a strong nonstationarity, such as spectral variation, which makes it difficult to use a constant wavelet for direct inversion in depth. To address this issue, we have developed a new wavelet extraction method by using a depth-wavenumber decomposition technique, which can generate depth variant wavelets to accommodate the nonstationarity of the depth-domain seismic data. The synthetic and real data applications have been used to test the effectiveness of our method. The directly inverted depth-domain AI indicates a good correlation with well-log data and a strong potential for reservoir characterization.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

An improved method to estimate Q based on the logarithmic spectrum of moving peak points

2019 ◽  
Vol 7 (2) ◽  
pp. T255-T263 ◽  
Author(s):  
Yanli Liu ◽  
Zhenchun Li ◽  
Guoquan Yang ◽  
Qiang Liu
Keyword(s):  
Seismic Waves ◽  
Wave Attenuation ◽  
Real Data ◽  
Peak Frequency ◽  
Seismic Reflection Data ◽  
The Real ◽  
Time Frequency ◽  
Reflection Data ◽  
Text Filtering ◽  
The Impact

The quality factor ([Formula: see text]) is an important parameter for measuring the attenuation of seismic waves. Reliable [Formula: see text] estimation and stable inverse [Formula: see text] filtering are expected to improve the resolution of seismic data and deep-layer energy. Many methods of estimating [Formula: see text] are based on an individual wavelet. However, it is difficult to extract the individual wavelet precisely from seismic reflection data. To avoid this problem, we have developed a method of directly estimating [Formula: see text] from reflection data. The core of the methodology is selecting the peak-frequency points to linear fit their logarithmic spectrum and time-frequency product. Then, we calculated [Formula: see text] according to the relationship between [Formula: see text] and the optimized slope. First, to get the peak frequency points at different times, we use the generalized S transform to produce the 2D high-precision time-frequency spectrum. According to the seismic wave attenuation mechanism, the logarithmic spectrum attenuates linearly with the product of frequency and time. Thus, the second step of the method is transforming a 2D spectrum into 1D by variable substitution. In the process of transformation, we only selected the peak frequency points to participate in the fitting process, which can reduce the impact of the interference on the spectrum. Third, we obtain the optimized slope by least-squares fitting. To demonstrate the reliability of our method, we applied it to a constant [Formula: see text] model and the real data of a work area. For the real data, we calculated the [Formula: see text] curve of the seismic trace near a well and we get the high-resolution section by using stable inverse [Formula: see text] filtering. The model and real data indicate that our method is effective and reliable for estimating the [Formula: see text] value.

Regularization and datuming of seismic data by weighted, damped least squares

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. U67-U76 ◽  
Author(s):  
Robert J. Ferguson
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Synthetic Data ◽  
Real Data ◽  
Structural Data ◽  
Irregular Sampling ◽  
Data Set ◽  
Near Surface ◽  
Damped Least Squares ◽  
Wavefield Extrapolation

The possibility of improving regularization/datuming of seismic data is investigated by treating wavefield extrapolation as an inversion problem. Weighted, damped least squares is then used to produce the regularized/datumed wavefield. Regularization/datuming is extremely costly because of computing the Hessian, so an efficient approximation is introduced. Approximation is achieved by computing a limited number of diagonals in the operators involved. Real and synthetic data examples demonstrate the utility of this approach. For synthetic data, regularization/datuming is demonstrated for large extrapolation distances using a highly irregular recording array. Without approximation, regularization/datuming returns a regularized wavefield with reduced operator artifacts when compared to a nonregularizing method such as generalized phase shift plus interpolation (PSPI). Approximate regularization/datuming returns a regularized wavefield for approximately two orders of magnitude less in cost; but it is dip limited, though in a controllable way, compared to the full method. The Foothills structural data set, a freely available data set from the Rocky Mountains of Canada, demonstrates application to real data. The data have highly irregular sampling along the shot coordinate, and they suffer from significant near-surface effects. Approximate regularization/datuming returns common receiver data that are superior in appearance compared to conventional datuming.

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