Unified elimination of 1D acoustic multiple reflection

Abstract

Migration, velocity and amplitude analysis are the employed processing steps to find the desired subsurface information from seismic reflection data. The presence of freesurface and internal multiples can mask the primary reflections for which many processing methods are built. The ability to separate primary from multiple reflections is desirable. Connecting Marchenko theory with classical one-dimensional inversion methods allows to understand the process of multiple reflection elimination as a datafiltering process. The filter is a fundamental wave field, defined as a pressure and particle velocity that satisfy the wave equation. The fundamental wave field does not depend on the presence or absence of free-surface multiples in the data. The backbone of the filtering process is that the fundamental wave field is computed from the measured pressure and particle velocity without additional information. Two different multiples-free datasets are obtained: either directly from the fundamental wave field or by applying the fundamental wave field to the data. In addition, the known schemes for Marchenko multiple elimination follow from the main equation. Numerical examples show that source and receiver ghosts, free-surface and internal multiples can be removed simultaneously using a conjugate gradient scheme. The advantage of the main equation is that the source wavelet does not need to be known and no preprocessing is required. The fact that the reflection coefficients can be obtained is an interesting feature that could lead to improved amplitude analysis and inversion than would be possible with other processing methods.

Marchenko Green's Function Retrieval in Layered Elastic Media from Two-Sided Reflection and Transmission Data

“…As can be learned from the black curve in Figure 3(b), matrix A has full rank, and hence can be inverted. When we apply singular value decomposition A = UΣV t and define the pseudo inverse as A ‡ = VΣ ‡ U t (where Σ ‡ contains the reciprocals of all non-zero singular values), we may now write F m = A ‡ B. Akin to the acoustic Marchenko problem, a range of alternative solvers might be used to compute the pseudo inverse [32,33].…”

On the retrieval of forward-scattered waveforms from acoustic reflection and transmission data with the Marchenko equation

“…the direct-wave approximation), the coda F U m can be resolved from ( 17) by linear inversion. A common strategy for the inversion is to rewrite the equation by a Neumann series expansion, which is guaranteed (at least in 1D for infinite frequency content) to converge as long as the spectral radius of operator R U is less than one [31], [34]. However, a variety of alternative numerical solvers might be employed [31], [34], [35], [36].…”

“…A common strategy for the inversion is to rewrite the equation by a Neumann series expansion, which is guaranteed (at least in 1D for infinite frequency content) to converge as long as the spectral radius of operator R U is less than one [31], [34]. However, a variety of alternative numerical solvers might be employed [31], [34], [35], [36]. For the construction of operator R U , we require access to a complete, well-sampled reflection response [37], [38] and sufficient aperture [39].…”

Marchenko Green’s Function Retrieval in Layered Elastic Media from Two-Sided Reflection and Transmission Data

“…As indicated by the black curve in Figure 3b, matrix A has full rank, and hence can be inverted. When we apply singular-value decomposition A = UΣV t and define the pseudo-inverse as A ‡ = VΣ ‡ U t (where Σ ‡ contains the reciprocals of all non-zero singular values), we may now write F m = A ‡ B. Akin to the acoustic Marchenko problem, a range of alternative solvers can be used to compute the pseudo-inverse [40,41].…”