Time-Domain Multidimensional Deconvolution: A Physically Reliable and Stable Preconditioned Implementation

Abstract

Multidimensional deconvolution constitutes an essential operation in a variety of geophysical scenarios at different scales ranging from reservoir to crustal, as it appears in applications such as surface multiple elimination, target-oriented redatuming, and interferometric body-wave retrieval just to name a few. Depending on the use case, active, microseismic, or teleseismic signals are used to reconstruct the broadband response that would have been recorded between two observation points as if one were a virtual source. Reconstructing such a response relies on the the solution of an ill-conditioned linear inverse problem sensitive to noise and artifacts due to incomplete acquisition, limited sources, and band-limited data. Typically, this inversion is performed in the Fourier domain where the inverse problem is solved per frequency via direct or iterative solvers. While this inversion is in theory meant to remove spurious events from cross-correlation gathers and to correct amplitudes, difficulties arise in the estimation of optimal regularization parameters, which are worsened by the fact they must be estimated at each frequency independently. Here we show the benefits of formulating the problem in the time domain and introduce a number of physical constraints that naturally drive the inversion towards a reduced set of stable, meaningful solutions. By exploiting reciprocity, time causality, and frequency-wavenumber locality a set of preconditioners are included at minimal additional cost as a way to alleviate the dependency on an optimal damping parameter to stabilize the inversion. With an interferometric redatuming example, we demonstrate how our time domain implementation successfully reconstructs the overburden-free reflection response beneath a complex salt body from noise-contaminated up- and down-going transmission responses at the target level.

Large-scale Marchenko imaging with distance-aware matrix reordering, tile low-rank compression, and mixed-precision computations

“…As far as the MDD process is concerned, our estimate is also compared with the directly modelled local reflectivity. This highlights how the error introduced by our algorithmic approximations is negligible when compared with the overall discrepancies in the estimated solution, due to the fact that we are solving a highly ill-posed Fredholm integral of the first kind (Vargas et al, 2021).…”

A Regularised Total Least Squares Approach for 1D Inverse Scattering

“…Then follows the solution of the GLM integral Equation (17). After the GLM kernel is recovered, the scattering potential, q, can be found using relation (16).…”

“…In addition, we refer to [23] for a discussion on a stability estimate of the Marchenko inversion where the bounds are on the scattering potential. In the application of recovering the scattering potential from scattering data, a pointwise estimate for the error is sufficient in view of relation (16). We denote…”

“…Unless stated otherwise, we use 10 iterations of the alternating method and 10 iterations of LSQR for the sub problems. The scattering potential is obtained by numerically differentiating the reconstructed kernel, as in (16).…”

“…In particular, the GLM approach with noisy data requires a total least squares (TLS) approach, and was studied numerically by [15]. Other regularised approaches for similar integral equations in seismic imaging are discussed in [16].…”