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

A comparison of Marchenko strategies for the treatment of internal multiples and their consequences for seismic imaging

“…We follow Santos et al (2020a) to give a brief overview of LSMME scheme in this section. To clarify our notation, the spatial coordinates are defined by their horizontal and depth components, for instance, x i = (x H , z i ), where x H are the horizontal coordinates and z i is the depth of an arbitrary boundary ∂ D i , such that the surface acquisition ∂ D 0 will be defined by x 0 = (x H , z 0 ).…”

“…In practice, R is obtained from deconvolution of R with the source time signature. The projected version of the revised Marchenko equations for the single-sided reflection response can be given by the following expressions (Zhang and Staring, 2018;Santos et al, 2020a):…”

“…To evaluate U − it is first necessary to obtain v + m . Santos et al (2020a) shows that v + m can be obtained if the Equation ( 1) is seen as an inverse problem, which can be solved using the least-squares scheme. So, starting with the affirmation presented in van der Neut and Wapenaar (2016), where is defined that {Rδ }(x 0 , x 0 ,t) = R(x 0 , x 0 ,t), we can rewrite these equations (1) in the following matricial form:…”

“…This study demonstrates that internal multiple reflections have influence on the velocity analysis at early processing stages and that MME can be seen as a new tool in the conventional pre-processing workflows of seismic data. In the same year, Santos et al (2020a) have proposed to formulate the application of the MME technique as a least-squares problem named of Marchenko multiples elimination by least-squares (LSMME) to avoid the stability issue related to the Neumann series expansion presented in the original formulation proposed by Zhang and Staring (2018). Staring et al (2018) developed an adaptive doublefocusing method (DFM) to remove the internal multiples caused by an overburden using source-receiver Marchenko redatuming.…”

“…As shown by Staring et al (2018), knowing the depth of the reservoir, it is possible to carry out a redatuming with the DFM for a region a little above it, thus allowing to obtain an image of the reservoir free of many of the multiple generated by layers above at a relatively low-cost. On the other hand, as shown in Santos et al (2020a), the use of LSMME allows to obtain a multiples-free image of all orders and from all structures in the medium, but paying a much higher computational cost than the DFM. As in Wapenaar et al (2021), which have discussed the different approaches to Marchenko redatuming, imaging and multiple elimination, using a common mathematical framework, in this paper, we compare the DFM with LSMME in the context of improving the image of a target region.…”