A least-squares based approach for the Marchenko internal multiple elimination scheme

Seismic images produced by migration of seismic data related to complex geologies, suchas pre-salt environments, are often contaminated by artifacts due to the presence of multipleinternal reflections. These reflections are created when the seismic wave is reflected morethan once in a source-receiver path and can be interpreted as the main coherent noise inseismic data. Several schemes have been developed to predict and subtract internal multiplereflections from measured data, such as the Marchenko multiple elimination (MME) scheme,which eliminates the referred events without requiring a subsurface model or an adaptivesubtraction approach. The MME scheme is data-driven, can remove or attenuate mostof these internal multiples, and was originally based on the Neumann series solution ofMarchenko’s projected equations. However, the Neumann series approximate solution isconditioned to a convergence criterion. In this work, we propose to formulate the MMEas a least-squares problem (LSMME) in such a way that it can provide an alternative thatavoids a convergence condition as required in the Neumann series approach. To demonstratethe LSMME scheme performance, we apply it to 2D numerical examples and compare theresults with those obtained by the conventional MME scheme. Additionally, we evaluatethe successful application of our method through the generation of in-depth seismic images,by applying the reverse-time migration (RTM) algorithm on the original data set and tothose obtained through MME and LSMME schemes. From the RTM results, we show thatthe application of both schemes on seismic data allows the construction of seismic imageswithout artifacts related to internal multiple events.

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Adaptive curvelet-domain primary-multiple separation

Geophysics ◽

10.1190/1.2904986 ◽

2008 ◽

Vol 73 (3) ◽

pp. A17-A21 ◽

Cited By ~ 43

Author(s):

Felix J. Herrmann ◽

Deli Wang ◽

Dirk J. (Eric) Verschuur

Keyword(s):

Least Squares ◽

High Frequency ◽

Real Data ◽

Adaptive Method ◽

Phase Errors ◽

Successful Separation ◽

Data Adaptive ◽

Offset Time ◽

Multiple Elimination

In many exploration areas, successful separation of primaries and multiples greatly determines the quality of seismic imaging. Despite major advances made by surface-related multiple elimination (SRME), amplitude errors in the predicted multiples remain a problem. When these errors vary for each type of multiple in different ways (as a function of offset, time, and dip), they pose a serious challenge for conventional least-squares matching and for the recently introduced separation by curvelet-domain thresholding. We propose a data-adaptive method that corrects amplitude errors, which vary smoothly as a function of location, scale (frequency band), and angle. With this method, the amplitudes can be corrected by an elementwise curvelet-domain scaling of the predicted multiples. We show that this scaling leads to successful estimation of primaries, despite amplitude, sign, timing, and phase errors in the predicted multiples. Our results on synthetic and real data show distinct improvements over conventional least-squares matching in terms of better suppression of multiple energy and high-frequency clutter and better recovery of estimated primaries.

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Efficient 2D multiple attenuation using SRME with curvelet-domain subtraction

Marine Geophysical Research ◽

10.1007/s11001-021-09464-8 ◽

2022 ◽

Vol 43 (1) ◽

Author(s):

Szu-Ying Lai ◽

Yunung Nina Lin ◽

Ho-Han Hsu

Keyword(s):

Finite Number ◽

Least Squares ◽

Seismic Data ◽

Parameter Determination ◽

Prediction Errors ◽

Geological Condition ◽

Multiple Model ◽

Multiple Attenuation ◽

Time Space ◽

Multiple Elimination

AbstractSurface Related Multiple Elimination (SRME) usually suffers the issue of either over-attenuation that damages the primaries or under-attenuation that leaves strong residual multiples. This dilemma happens commonly when SRME is combined with least-squares subtraction. Here we introduce a more sophisticated subtraction approach that facilitates better separation of multiples from primaries. Curvelet-domain subtraction transforms both the data and the multiple model into the curvelet domain, where different frequency bands (scales) and event directions (orientations) are represented by a finite number of curvelet coefficients. When combined with adaptive subtraction in the time–space domain, this method can handle model prediction errors to achieve effective subtraction. We demonstrate this method on two 2D surveys from the TAiwan Integrated GEodynamics Research (TAIGER) project. With a careful parameter determination flow, our result shows curvelet-domain subtraction outperforms least-squares subtraction in all geological settings. We also present one failed case where specific geological condition hinders proper multiple subtraction. We further demonstrate that even for data acquired with short cables, curvelet-domain subtraction can still provide better results than least-squares subtraction. We recommend this method as the standard processing flow for multi-channel seismic data.

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Implementation of the Marchenko Multiple Elimination algorithm

Geophysics ◽

10.1190/geo2020-0196.1 ◽

2020 ◽

pp. 1-54

Author(s):

Jan Thorbecke ◽

Lele Zhang ◽

Kees Wapenaar ◽

Evert Slob

Keyword(s):

Time Domain ◽

Neumann Series ◽

Transmission Losses ◽

Multiple Reflections ◽

Elimination Algorithm ◽

Domain Truncation ◽

Internal Multiples ◽

The Time Domain ◽

Multiple Elimination ◽

Elimination Scheme

The Marchenko multiple elimination and transmission compensation schemes retrieve primary reflections in the two-way traveltime domain without model information or using adaptive subtraction. Both schemes are derived from projected Marchenko equations and similar to each other, but use different time-domain truncation operators. The Marchenko multiple elimination scheme retrieves a new dataset without internal multiple reflections. The transmission compensated Marchenko multiple elimination scheme does the same and additionally compensates for transmission losses in the primary reflections. Both schemes can be solved with an iterative algorithm based on a Neumann series. At each iteration, a convolution or correlation between the projected focusing function and the measured reflection response are performed and after each convolution or correlation, a truncation in the time domain is applied. After convergence, the resulting projected focusing function is used for retrieving the transmission compensated primary reflections and the projected Green’s function is used for the physical primary reflections. We demonstrate that internal multiples are removed by using time-windowed input data that only contain primary reflections. We evaluate both schemes in detail and develop an iterative implementation that reproduces the presented numerical examples. The software is part of our open-source suite of programs and fits into the Seismic Unix software suite of the Colorado School of Mines.

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Estimation of an Overidentified Equation in an Econometric Model by the Method of Generalized Least Squares

Communications in Statistics - Simulation and Computation ◽

10.1080/03610917308548419 ◽

1973 ◽

Vol 1 (2) ◽

pp. 167-181

Author(s):

Monique Chakravarti-Kelm

Keyword(s):

Least Squares ◽

Econometric Model ◽

Generalized Least Squares

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A Note on Rao's Unified Theory of Least Squares

Communications in Statistics - Simulation and Computation ◽

10.1080/03610917408548485 ◽

1974 ◽

Vol 3 (9) ◽

pp. 909-912

Author(s):

C. W. Chen

Keyword(s):

Least Squares ◽

Unified Theory

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Large Sample and Jackknife Procedures for Small Sample Orthogonal Least Squares Inference

Communications in Statistics - Simulation and Computation ◽

10.1080/03610917508548348 ◽

1975 ◽

Vol 4 (2) ◽

pp. 193-202

Author(s):

Peter Anderson

Keyword(s):

Least Squares ◽

Small Sample ◽

Large Sample ◽

Orthogonal Least Squares

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Heteroscedastic Methods for Performing All Pairwise Comparisons of Regression Lines Associated With J Independent Groups

Methodology ◽

10.1027/1614-2241/a000097 ◽

2015 ◽

Vol 11 (3) ◽

pp. 110-115 ◽

Cited By ~ 2

Author(s):

Rand R. Wilcox ◽

Jinxia Ma

Keyword(s):

Least Squares ◽

Pairwise Comparisons ◽

Simple Extension ◽

Type I ◽

Type I Errors ◽

Well Elderly ◽

Using Data

Abstract. The paper compares methods that allow both within group and between group heteroscedasticity when performing all pairwise comparisons of the least squares lines associated with J independent groups. The methods are based on simple extension of results derived by Johansen (1980) and Welch (1938) in conjunction with the HC3 and HC4 estimators. The probability of one or more Type I errors is controlled using the improvement on the Bonferroni method derived by Hochberg (1988) . Results are illustrated using data from the Well Elderly 2 study, which motivated this paper.

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