An extension of least-squares redatuming: Simultaneous reconstruction of overburden reflectivities and virtual data

Highlights

• A model-driven redatuming approach using constrained least-squares inversion.

• Simultaneous updating overburden reflectivities and virtual data on the datum.

• Applications of least-squares redatuming to synthetic and offshore seismic data.

Abstract

In a controlled-source seismology scenario, scattered wavefields from deep targets, e.g., the petroleum reservoirs, are often distorted by strong heterogeneities of the overburden media, complex topography and irregular acquisition. It is key for target-oriented seismic imaging to accurately reconstruct the scattering wavefields from the deep structures. Recent studies have promoted seismic redatuming for removing the imprint of the overburden through reconstructing virtual data just above the target area. Among the model-based redatuming approaches, least-squares inversion has advantages in mitigating the artifacts due to limited recording aperture and irregular spatial sampling, and allowing for the introduction of appropriate constraints to the reconstructed data. However, conventional least-squares redatuming (LSR) suffers from the spurious events and noises associated with the wavefield interaction of the overburden reflectivities close to the datum. To overcome this problem, we establish an efficient double-parameter LSR method that simultaneously updates the overburden reflectivities and the virtual data through a constrained optimization. Synthetic and field data examples demonstrate that this extended LSR method can effectively retrieve the scattering wavefields from the deep part and improve the target-oriented prestack depth migration below the datum.

Introduction

Redatuming is an operation applied to seismic data that translates the positions of sources or/and receivers and thus makes the reconstructed seismic traces appear to be acquired with virtual sources or/and receivers at a new datum [5], [40]. It can mitigate the effects of irregular topography and complex geological structures in the overburden such as strong heterogeneities [3], [45]. As a result, the redatumed data can be used to apply an imaging process where the wave propagation and scattering above the datum level can be completely neglected (e.g., [47]). Moreover, this can naturally lead to a target-oriented approach that alleviates the computational burden associated with least-squares migration (LSM) and full waveform inversion (FWI) by limiting the inversion process to only a portion of the subsurface where an accurate and high-resolution characterization is needed, for instance at reservoir level, e.g., Yuan et al. [46], Cui et al. [10], Guo and Alkhalifah [14].

Seismic redatuming generally requires an estimate of the transmission response of the overburden, which can either be computed with a macrovelocity model of the subsurface or retrieved from the recorded data. This leads to two types of methodology: model-based and correlation-based [31]. On the one hand, given a macrovelocity model, the acquired data can be extrapolated to the new datum with a Kirchhoff integral or wave-equation continuation operator [6], [20], [23], [2], [12], [45]. The model-based redatuming approaches effectively correct the seismic traces using traveltimes determined by the macrovelocity model. On the other hand, when receivers are deployed below the overburden (e.g., in a deviated or horizontal well) to measure the transmission response directly, correlation-based redatuming can be applied to construct virtual source/receiver gathers, without any prior knowledge of the medium [9], [30], [4], [38]. It requires applying weighted correlations of the traces with each other, followed by summation over all sources (or sometimes receivers), and thus use natural-phase information in the data to time shift the weighted seismic traces so that they appear to be excited by sources (or recorded by receivers) on a new datum. A theoretical overview of model- and correlation-based redatuming methods has been presented by Schuster and Zhou [31]. If buried receivers are not available, novel approaches based on the Marchenko method for wavefield focusing have been presented for redatuming of single-sided data. In this context, the Green's function is iteratively retrieved using only the reflection responses of the medium and an estimate of the first arrival at the recording surface from the virtual source [29], [7], [42], [27]. Alternatively, internal scattering can be imaged with the background Green's function from the surface to each subsurface location in a generalized internal multiple imaging procedure [48], [1]. More or less, all these methods are affected by recording aperture and spatial sampling.

In theory, redatuming can be formulated as an inverse modelling, in which the virtual data on the datum constitutes the model parameter to be estimated. Because the acquisition aperture is finite and the spatial sampling may be irregular, conventional redatuming methods only form an approximate inverse and produce many artifacts in the solution space. Ji [17] and Ferguson [13] suggested to suppress the artifacts and improve the reconstruction accuracy of wavefield extrapolation-based redatuming [6] through a least-squares optimization process. Accordingly, Xue and Schuster [43] implemented a least-squares interferometric redatuming scheme in the context of vertical seismic profile (VSP) experiment. Alternatively, Wapenaar and van der Neut [41] extended redatuming with a multidimensional deconvolution (MDD) based on point-spread functions (PSFs). Recently new representations using PSFs provide adaptation of the iterative Marchenko scheme for imperfectly sampled data [16].

Redatuming aims to completely remove the effects of a complex overburden. Even if surface and internal multiples are ignored (e.g., for deep datum levels) for the moment, one must properly account for transmission and primary reflections in the overburden media. The inaccuracy of the overburden macrovelocity model prevents a reasonable reconstruction of the virtual data. By formulating the redatuming operation with an extended Born representation, Guo and Alkhalifah [15] implemented a simultaneous inversion of the overburden long-wavelength velocity model and the virtual data at the datum. In fact, the overburden short-wavelength velocity structures (or reflectivities) could affect model-based least-squares redatuming (LSR). As we will demonstrate in this paper, imprints of the overburden reflections or/and diffractions generated very close to the datum could hinder the correct measurement of the data misfit and impede the iterative inversion. In the presence of complex and irregular geologic bodies (e.g., salt body and igneous rock) just above or cross the datum, removing these overburden reflections and diffractions, for instance with a delicate time window (e.g., [42]), is not trivial. To address this problem, we present a simultaneous inversion for the overburden reflectivities and the virtual data in LSR, given the overburden macrovelocity model as a prior.

The structure of this paper is given as follows: First, we describe the forward problem of this double-parameter LSR based on the convolution-type reciprocity theorem and the first-order scattering theory. Then, instead of solving the normal equation that explicitly incorporate the inverse Hessian, we simultaneously update the two classes of model parameter with a gradient-type iterative inversion based on the L2-norm misfit function. In this context, the functional gradients are derived using the adjoint-state method and constructed through correlation of the forward and adjoint wavefields. To enhance the robustness and suppress the numerical artifacts, we apply sparsity constraints on the overburden reflectivities and the virtual data using a split Bregman method. Finally, synthetic and field data examples are presented to demonstrate the effectiveness of the proposed approach to tackle irregular sampling and overburden effects.