A Marchenko Multiple Elimination Solution Based on the Beyond Neumann Method

A Comparison Between Different Solutions for the Marchenko Multiple Eliminations Scheme

“…The schemes MME, LSMME, and BNMME are based on the projected Marchenko equations for the single-sided reflection response [8][9][10]:…”

“…The Θ is a truncation operator to exclude values outside the window (ε, t -ε), in which ε is a positive value to account for the finite bandwidth. R and R* are multidimensional convolution and correlation operators [1,8]. Following Zhang and Slob [4], Equation (1) must be evaluated for each instant time t2 and their value collected to be stored in a new function containing only primary reflections as:…”

“…The BNMME scheme is based on solving the linear system in Equation ( 6) using the beyond Neumann method [12]. In this approach, the solution is obtained using the following recursive scheme: (7) in which αk represents a relaxation parameter, which is updated according to (8) with .…”

“…The MME solution proposed by Zhang and colleagues [2] is based on the Neumann series approximation (NEMME), so recently, Santos and colleagues [1] have proposed to formulate the MME scheme as a least-squares problem (LSMME), which averts the convergence criterion of the Neumann series approximation, and evaluates this approach in a complex 2D synthetic numerical example. Subsequently, to reduce the MME empirical scale factor dependence, Santos and colleagues [8] proposed an alternative solution based on the beyond Neumann scheme (BNMME) and showed that BNMME is more suitable in situations where it is difficult to obtain an ideal scale factor. The experiments developed by Zhang and colleagues [2] and Santos and colleagues [1,8] showed that when seismic data have previously gone through a high-quality pre-processing stage, i.e., deghosting, removal of free-surface multiples, and deconvolution with an estimated source wavelet, the schemes NEMME, LSMME, and BNMME successfully eliminates or attenuates multiple internal reflections.…”

“…Subsequently, to reduce the MME empirical scale factor dependence, Santos and colleagues [8] proposed an alternative solution based on the beyond Neumann scheme (BNMME) and showed that BNMME is more suitable in situations where it is difficult to obtain an ideal scale factor. The experiments developed by Zhang and colleagues [2] and Santos and colleagues [1,8] showed that when seismic data have previously gone through a high-quality pre-processing stage, i.e., deghosting, removal of free-surface multiples, and deconvolution with an estimated source wavelet, the schemes NEMME, LSMME, and BNMME successfully eliminates or attenuates multiple internal reflections. However, despite the experiments showing the power of the referred schemes in attenuating the internal multiples, their computational performance still needs to be evaluated.…”