Comparison of the ℓ1 and ℓ2 norms applied to one‐at‐a‐time spike extraction from seismic traces

Geophysics ◽  
1984 ◽  
Vol 49 (11) ◽  
pp. 2048-2052 ◽  
Author(s):  
I. Barrodale ◽  
C. A. Zala ◽  
N. R. Chapman
Keyword(s):  
Spike Train ◽  
Least Squares ◽  
Seismic Data ◽  
The Third ◽  
Different Characteristics

We present an algorithm for deconvolving a seismic trace by extracting spikes one at a time, thereby obtaining a sparsely populated spike train. Three versions of this algorithm are then compared empirically, by applying them to several examples of synthetic and real seismic data. The first two versions correspond to the use of the [Formula: see text] (least‐absolute‐values) and [Formula: see text] (least‐squares) norms, while the third is a faster and more compact version of the [Formula: see text], algorithm. The [Formula: see text] procedures are shown to exhibit different characteristics which are often desirable, and the results are generally superior to those of the [Formula: see text] procedure for one‐at‐a‐time spike extraction. The use of the fast [Formula: see text] algorithm is advocated in practice for efficient and effective deconvolution.

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.

Comparison of plasma progesterone profiles in cyclic, pregnant, pseudopregnant and hysterectomized pigs between 8 and 27 days after oestrus

1988 ◽  
Vol 119 (1) ◽  
pp. 111-116 ◽  
Author(s):  
G. J. King ◽  
R. Rajamahendran
Keyword(s):  
Least Squares ◽  
Analysis Of Variance ◽  
The Other ◽  
Corpora Lutea ◽  
Plasma Progesterone ◽  
The Third ◽  
Secretory Potential ◽  
The Mean

ABSTRACT Plasma progesterone concentrations were compared in cyclic (n = 12), pregnant (n =12), oestradiol-induced pseudopregnant (n=12) and hysterectomized gilts (n=10) between days 8 and 27 after oestrus. The results were grouped into periods covering days 8–13, 14–20 and 21–27 and analysed by least-squares analysis of variance. Plasma progesterone concentrations were significantly (P<0·001) higher in hysterectomized compared with other groups between days 8 and 13. Progesterone concentrations declined rapidly after day 14 in cyclic females and gradually in the other groups. Throughout the third and fourth weeks the mean progesterone concentrations for hysterectomized animals were consistently higher than for pseudopregnant animals (P<0·05). The pregnant group means were below but not significantly different from the hysterectomized means in both of the last two periods. The greater progesterone concentrations in hysterectomized gilts indicated that secretion is high without any conceptus-produced or -mediated luteotrophin, and corpora lutea in cyclic, pregnant or pseudopregnant gilts may never reach full secretory potential. J. Endocr. (1988) 119, 111–116

scholarly journals Improving adaptive subtraction in seismic multiple attenuation

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. V59-V67 ◽  
Author(s):  
Shoudong Huo ◽  
Yanghua Wang
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Primary Energy ◽  
Multiple Models ◽  
Multiple Model ◽  
Data Set ◽  
Multiple Attenuation ◽  
Matching Filter ◽  
Theoretical Analyses ◽  
Different Order

In seismic multiple attenuation, once the multiple models have been built, the effectiveness of the processing depends on the subtraction step. Usually the primary energy is partially attenuated during the adaptive subtraction if an [Formula: see text]-norm matching filter is used to solve a least-squares problem. The expanded multichannel matching (EMCM) filter generally is effective, but conservative parameters adopted to preserve the primary could lead to some remaining multiples. We have managed to improve the multiple attenuation result through an iterative application of the EMCM filter to accumulate the effect of subtraction. A Butterworth-type masking filter based on the multiple model can be used to preserve most of the primary energy prior to subtraction, and then subtraction can be performed on the remaining part to better suppress the multiples without affecting the primaries. Meanwhile, subtraction can be performed according to the orders of the multiples, as a single subtraction window usually covers different-order multiples with different amplitudes. Theoretical analyses, and synthetic and real seismic data set demonstrations, proved that a combination of these three strategies is effective in improving the adaptive subtraction during seismic multiple attenuation.

Prestack imaging of seismic data using Lp iterative reweighted least-squares wavefield extrapolation filters in the frequency-space domain

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. V243-V252
Author(s):  
Wail A. Mousa
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Least Squares Method ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Weighted Least Squares ◽  
Data Set ◽  
Frequency Space ◽  
Wavefield Extrapolation ◽  
Iterative Reweighted Least Squares

A stable explicit depth wavefield extrapolation is obtained using [Formula: see text] iterative reweighted least-squares (IRLS) frequency-space ([Formula: see text]-[Formula: see text]) finite-impulse response digital filters. The problem of designing such filters to obtain stable images of challenging seismic data is formulated as an [Formula: see text] IRLS minimization. Prestack depth imaging of the challenging Marmousi model data set was then performed using the explicit depth wavefield extrapolation with the proposed [Formula: see text] IRLS-based algorithm. Considering the extrapolation filter design accuracy, the [Formula: see text] IRLS minimization method resulted in an image with higher quality when compared with the weighted least-squares method. The method can, therefore, be used to design high-accuracy extrapolation filters.

Least-squares data-to-data migration: An approach for migrating free-surface-related multiples

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. S83-S94 ◽  
Author(s):  
Yikang Zheng ◽  
Yibo Wang ◽  
Xu Chang
Keyword(s):  
Free Surface ◽  
Least Squares ◽  
Seismic Data ◽  
Signal To Noise Ratio ◽  
Data Migration ◽  
Subsurface Imaging ◽  
Full Data ◽  
Recording Time ◽  
Numerical Tests ◽  
Marine Seismic

Free-surface-related multiples can provide extra illumination of the subsurface and thus can be usefully included in migration procedures. However, most multiple migration approaches require separation of primaries and free-surface-related multiples or at least prediction of multiples in advance, which is time consuming and prone to errors. The data-to-data migration (DDM) method migrates free-surface-related multiples by forward and backward propagating the recorded full data (containing primaries and free-surface-related multiples). For DDM, there is no need to predict or separate multiples, but the migration results suffer from the crosstalk generated by crosscorrelations of undesired seismic events, e.g., primaries and second-order free-surface-related multiples. We have developed least-squares DDM (LSDDM) for marine data to eliminate the crosstalk generated by DDM. In each iteration, the forward-propagated primaries and free-surface-related multiples are crosscorrelated with the backward-propagated primary and free-surface-related multiple residuals to form the reflectivity gradient. We use a three-layer model and the Marmousi model for numerical tests. The results validate that LSDDM can provide a migrated image with higher signal-to-noise ratio and more balanced amplitudes than DDM. The LSDDM approach might be valuable for general subsurface imaging for marine seismic data when the migration velocity is accurate, and the acquired data have sufficient recording time.

Target-oriented wave-equation inversion

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. A35-A38 ◽  
Author(s):  
Alejandro A. Valenciano ◽  
Biondo Biondi ◽  
Antoine Guitton
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Seismic Data ◽  
Hessian Matrix ◽  
Velocity Model ◽  
Complex Model ◽  
Inverse Filtering ◽  
Band Limited ◽  
Filtering Problem ◽  
Number Of Iterations

A target-oriented strategy can be applied to estimate a wave-equation least-squares inverse (LSI) image. By explicitly computing the wave-equation Hessian, the LSI image is obtained as the solution of a nonstationary least-squares inverse filtering problem. The rows of the Hessian are the nonstationary filters containing information about the acquisition geometry, the velocity model, and the band-limited characteristics of the seismic data. By exploiting the sparsity and the structure of the Hessian matrix, a large number of iterations, necessary to achieve convergence, can be computed cheaply. The results on a structurally complex model show the improvements of the LSI image versus the migrated image.

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