Reflection waveform inversion (RWI) is a technique that uses pure reflection data to estimate subsurface background velocity, relying on evolving seismic images. Conventional RWI operates in a cyclic workflow, with two key components in each cycle—migration and reflection tomography. Conventional RWI may result in suboptimal background velocity estimation, partly due to limited or unresolved resolution within each component in each cycle. While gradient pre-conditioning with the reciprocal of Hessian information helps resolve this issue in both components of RWI, it becomes impractical for a large number of model parameters. One-way reflection waveform inversion (ORWI) is a reflection waveform inversion technique in which the forward modelling scheme operates in one direction (downward and then upward) via virtual parallel depth levels within the medium. Leveraging the ORWI framework, we decompose and reduce the linear Hessian operator (also known as the approximate Hessian or Gauss–Newton Hessian) into multiple smaller suboperators. In particular, the diagonal blocks of the monofrequency approximate Hessian operators, each corresponding to a single depth level within the medium, are extracted and inverted to pre-condition the corresponding monofrequency gradients in both the migration and reflection tomography components of ORWI. This depth-dependent gradient pre-conditioning transforms standard ORWI into a high-resolution, yet computationally feasible version aimed at addressing suboptimal velocity estimation, referred to as high-resolution ORWI. The effectiveness of the proposed approach is demonstrated through successful applications to synthetic data examples.

Conventional reflection waveform inversion solves a two-parameter seismic inverse problem alternately for subsurface reflectivity and acoustic background velocity as the model parameters. It seeks to reconstruct a low-wavenumber velocity model of the subsurface from pure reflection data cyclically, through alternating migration and tomography loops, such that the remodelled data fits the observed data. Low-resolution seismic images with unpreserved amplitudes, full-wave inconsistency in the short-offset data and cycle skipping in the long-offset are perceived as the main reasons for suboptimal tomographic updates and slow convergence in conventional reflection waveform inversion. In the context of one-way reflection waveform inversion, this paper addresses the listed limitations through four main components. First, it augments one-way reflection waveform inversion with a computationally affordable preconditioned least-squares wave equation migration algorithm to ensure high-resolution reflectors with preserved amplitudes. Second, the paper verifies how well the full-wave consistency condition in the short-offset data is satisfied in one-way reflection waveform inversion and suggests muting inconsistent short-offset residual waveforms in the tomography loop to attenuate their adverse imprint. Third, the paper suggests extending the migration offset beyond short offsets to improve both the illumination and the signal-to-noise ratio of the reflectors. Fourth, the paper presents a data-selection algorithm to exclude the damaging effect of the cycle-skipped long-offset data in the tomography loop. The effectiveness of the proposed one-way reflection waveform inversion algorithm is finally validated through three numerical examples, demonstrating its capability to recover high-fidelity tomograms.

Since the appearance of wave-equation migration, many have tried to improve the resolution and effectiveness of this technology. Least-squares wave-equation migration is one of those attempts that tries to fill the gap between the migration assumptions and reality in an iterative manner. However, these iterations do not come cheap. A proven solution to limit the number of least-squares iterations is to correct the gradient direction within each iteration via the action of a preconditioner that approximates the Hessian inverse. However, the Hessian computation, or even the Hessian approximation computation, in large-scale seismic imaging problems involves an expensive computational bottleneck, making it unfeasible. Therefore, we propose an efficient computation of the Hessian approximation operator, in the context of one-way wave-equation migration (WEM) in the space-frequency domain. We build the Hessian approximation operator depth by depth, considerably reducing the operator size each time it is calculated. We prove the validity of our proposed method with two numerical examples. We then extend our proposal to the framework of full-wavefield migration, which is based on WEM principles but includes interbed multiples. Finally, this efficient preconditioned least-squares full-wavefield migration is successfully applied to a dataset with strong interbed multiple scattering.

Since seismic imaging creates an image of the subsurface structure based on information received from the measured wavefield, it is essential to fully utilize the reflected waves. Full Wavefield Modeling (FWMod) was developed with recursive and iterative up-and-down wavefield propagation, using one-way wave propagation, to model both primary and multiple reflections. Using FWMod as the modeling engine, Full Wavefield Migration (FWM) has been introduced to directly image data including internal multiples, where internal multiple crosstalk is suppressed automatically via an inversion-based data-fitting process. This avoids the need for applying internal multiple removal, which is often challenging. Conventional one-way wave propagators calculated in the wavenumber domain, like the phase shift (PS) operator, have limitations when applied to strongly inhomogeneous media. Even when computing a new operator at each lateral grid point, they still suffer difficulties because the medium is assumed to be locally homogeneous. In the past, matrix eigendecomposition has been proposed as a way to create accurate, local velocity-based one-way propagation operators. In this article, an accurate propagator based on eigendecomposition is incorporated into FWMod and FWM. In the numerical examples, four models with strong lateral velocity variations were used to test the propagator. With a comparison of the conventional FWM based on the PS operator with input data including FWMod and a finite-difference (FD) approach, the numerical examples demonstrated that the proposed method has the potential to significantly enhance image reflectivity, suppress internal multiples, and maintain convergence speed during the least-squares inversion.

Conventional full waveform inversion has proven ineffective in recovering background velocity models for the targets out of the reach of refracted and diving waves. Therefore, different reflection waveform inversion (RWI) tools have been presented so far, one of which is joint migration inversion (JMI). JMI, unlike most similar tools, is parameterized by two classes of parameters, reflectivity and velocity, and equipped with decoupled imaging and tomographic sensitivity kernels. The fundamental characteristic shared by JMI or any RWI technique is the arrival-time consistency between the anchor parts of the modeled and observed data. This characteristic helps and keeps any reflection waveform tomography tool robust against cycle-skipping in near-offset data. In this paper, through examining the tomographic gradient of JMI, we show that conventional JMI suffers from cycle skipping in near-offset traces, which degenerates the fidelity of the tomographic gradient of JMI. Next, we demonstrate how the degeneracy can be lifted from two different perspectives, either by muting the misguiding tomographic wavepaths or migration isochrones built by near-offset residuals. Coupled with this, we show how excluding the cycle-skipped far-offset data significantly improves the validity of the tomographic gradient of JMI.