The accuracy of a model obtained by full-waveform inversion can be estimated by analysing the sensitivity of the data to perturbations of the model parameters in selected subsurface points. Each perturbation requires the computation of the seismic response in the form of Born scattering data for a typically very large number of shots, making the method time consuming. The computational cost can be significantly reduced by considering the point where the subsurface parameters are perturbed as a Born scatterer. Instead of modelling each shot separately, reciprocity relations provide the Green functions from the sources to the scatterer in terms of Green’s functions from the scatterer to the sources. In this way, the Born scattering data from a single point in the isotropic elastic case for a marine acquisition with pressure sources and receivers can be expressed in terms of the Green functions for force and moment tensor sources located at the scatterer and only a small number of forward runs are required. A 2-D example illustrates how the result can be used to determine the hessian and local covariance matrix for the model parameters at the scatterer at the cost of 5 forward simulation. ... Read more

Recently, the focus of reflection seismologists has shifted to applications where a high-resolution image of the subsurface is required. Least-Squares Reverse-Time Migration (LSRTM) is a common tool used to compute such images. Still, its high computational costs have led seismologists to use target-oriented LSRTM for imaging only a small target of interest within a larger subsurface block. Redatuming the data to the upper boundary of the target of interest is one approach to target-oriented LSRTM. Still, many redatuming methods cannot account for multiple scattering within the overburden. We apply a target-oriented least-squares reverse time migration algorithm that integrates Marchenko redatuming and double-focusing to a field dataset. This redatuming method accounts for all orders of multiple scattering in the overburden, thus improving the accuracy of target-oriented LSRTM. Moreover, we demonstrate the effectiveness of a double-focusing algorithm in reducing the data size by decreasing both spatial and temporal dimensions of the model and the data. The algorithm's performance is evaluated using field data acquired in the Norwegian Sea. The numerical results show that our target-oriented LSRTM algorithm can reduce the internal multiple effects and increase the resolution of the resulting image. ... Read more

Least-squares reverse-time migration (LSRTM) is a method that seismologists utilize to compute a high-resolution subsurface image. Nevertheless, LSRTM is a computationally demanding problem. One way to reduce the computational costs of the LSRTM is to choose a small region of interest (ROI) and compute the image of that region. However, finding representations that account for the wavefields entering the target region from the surrounding boundaries is necessary. This article confines the ROI between two boundaries above and below this region. The acoustic reciprocity theorem is employed to derive representations for the wavefields at the upper and lower boundaries of the target region. With the help of these representations, a target-enclosed LSRTM algorithm is developed to compute a high-resolution image of the ROI. Moreover, the possibility of using virtual receivers created by Marchenko redatuming is investigated. ... Read more

Reflection seismology aims to estimate the Earth's subsurface elastic parameters for further investigation by geologists and engineers. This involves generating elastic waves using seismic sources and recording the Earth's response with receivers. The subsurface model is typically considered a combination of a background model and a short-wavelength reflectivity model. There are two main paths to estimate these parameters: non-linear waveform inversion to directly compute the elastic parameters or depth migration to estimate a structural image or reflectivity of the subsurface.

Reverse-Time Migration (RTM) is a common depth migration technique that migrates recorded wavefields from the space-time domain to the space-depth domain. It utilizes the Born approximation and the adjoint of the Born operator to produce an RTM image. However, RTM can suffer from errors, such as noise, temporal and spatial limitations, and multiple reflections.

Least-Squares Reverse-Time Migration (LSRTM) is used to overcome some of these errors. LSRTM involves resolving the reflectivity model by least-squares inversion, which is computationally expensive. Gradient-based optimization algorithms are often employed to reduce the computational burden, but they still require solving the wave equation and its adjoint for a large model in multiple iterations. One way to reduce the computational cost is by limiting the computational domain to a target region of interest.

Target-oriented LSRTM, known as TOLSRTM, focuses on the wavefield just above the target by bypassing the overburden. This approach proves beneficial when the overburden generates strong internal multiple reflections that obscure the reflections from the target area. However, a redatuming method is required to predict all orders of multiples. Marchenko redatuming is a data-driven technique that predicts the Green's functions at the boundary of the target region, incorporating all orders of internal multiples. It allows for double-sided redatuming, considering both the source and receiver perspectives. By combining the LSRTM algorithm and Marchenko double-focusing, a target-oriented LSRTM algorithm is devised that can predict interactions between the target and overburden and remove the effects of the overburden in the image. Predicting these interactions results in an artifact-free image, a better convergence rate, and a high-resolution image of the target.

Target-oriented migration algorithms typically consider only the upper horizontal boundary of the region of interest (ROI), neglecting wavefields entering the ROI from the medium beneath the lower boundary. To address this, a target-enclosed LSRTM algorithm is proposed, including both the ROI's upper and lower boundaries. Including the lower boundary provides transmission information and can improve inversion convergence. In addition, this algorithm is adopted for virtual receivers created by Marchenko redatuming. In the case of physical receivers at the boundaries of the target zone, the target-enclosed algorithm can incorporate the transmission information emanating from the lower boundary to the upper one. Consequently, when the initial model is far from the actual model, the resulting image partly recovers the long wavelength part of the model in agreement with the Born approximation criteria. Moreover, when an initial model closer to the actual model is used, the algorithm can partially recover the vertical interfaces of the perturbation. In the case of virtual receivers at the boundaries of the target zone, since the Marchenko redatuming is performed in the initial background model, the redatumed wavefields at the lower boundary suffer from kinematic errors. Therefore, the algorithm can not recover the long wavelength part of the model.

The thesis concludes with a discussion of the results obtained from applying the algorithms to marine datasets. The images resulting from the Marchenko double-focusing based target-oriented LSRTM algorithm show improvements in both resolution and artifact reduction by suppressing the overburden generated internal multiple effects. Moreover, the double-focusing enables the user to reduce the computational costs of the LSRTM algorithm and choose finer spatial sampling for the image.

An appendix proposes a formulation for integrating the target-oriented algorithms with non-linear inversion like Full Waveform Inversion (FWI). The results of this proposed algorithm show its effectiveness by reducing the internal multiple related artifacts and increasing resolution and faster convergence. ... Read more

Geophysicists have widely used least-squares reverse-time migration (LSRTM) to obtain high-resolution images of the subsurface. However, LSRTM is computationally expensive and it can suffer from multiple reflections. Recently, a target-oriented approach to LSRTM has been proposed, which focuses the wavefield above the target of interest. Remarkably, this approach can be helpful for imaging below complex overburdens and subsalt domains. Moreover, this approach can significantly reduce the computational burden of the problem by limiting the computational domain to a smaller area. Nevertheless, target-oriented LSRTM still needs an accurate velocity model of the overburden to focus the wavefield accurately and predict internal multiple reflections correctly. A viable alternative to an accurate velocity model for internal multiple prediction is Marchenko redatuming. This method is a novel data-driven method that can predict Green's functions at any arbitrary depth, including all orders of multiples. The only requirement for this method is a smooth background velocity model of the overburden. Moreover, with Marchenko double-focusing, one can make virtual sources and receivers at a boundary above the target and bypass the overburden. This paper proposes a new algorithm for target-oriented LSRTM, which fits the Marchenko double-focused data with predicted data. The predicted data of the proposed method is modelled by a virtual source term created by Marchenko double-focusing on a boundary above the target of interest. This virtual source term includes all the interactions between the target and the overburden. Moreover, the Marchenko double-focused data and the virtual source term are free of multiples generated in the overburden. Consequently, our target-oriented LSRTM algorithm suppresses the multiples purely generated inside the overburden. Our algorithm correctly accounts for all orders of multiples caused by the interactions between the target and the overburden, resulting in a significant reduction of the artefacts caused by the overburden internal multiple reflections and improves amplitude recovery in the target image compared to conventional LSRTM. ... Read more

Least-squares reverse time migration (LSRTM) is a common imaging technique that geophysicists have been using to obtain high-resolution images. Nevertheless, the high computational cost shifted the focus of researchers to the target-oriented approach. In this approach, by limiting the computational grid to a relatively smaller region, the computational cost of the LSRTM is significantly reduced. However, without an accurate model of the overburden, which can model all orders of overburden-generated multiples, the image produced by this approach suffers from overburden-related artifacts. Recently, Marchenko double-focusing presented itself as a powerful data-driven tool that can focus the recorded wavefield above the target region and eliminate the effects of the overburden-related multiple reflections. This paper proposes a forward modeling and inversion algorithm based on Marchenko double-focusing for target-oriented LSRTM to produce artifact-free high-resolution images. ... Read more

The Hessian matrix plays an important role in correct interpretation of the multiple scattered wave fields inside the FWI frame work. Due to the high computational costs, the computation of the Hessian matrix is not feasible. Consequently, FWI produces overburden related artifacts inside the target zone model, due to the lack of the exact Hessian matrix. We have shown here that Marchenko-based target-oriented Full Waveform Inversion can compensate the need of Hessian matrix inversion by reducing the nonlinearity due to overburden effects. This is achieved by exploiting Marchenko-based target replacement to remove the overburden response and its interactions with the target zone from residuals and inserting the response of the updated target zone into the response of the entire medium. We have also shown that this method is more robust with respect to prior information than the standard gradient FWI. Similarly to standard Marchenko imaging, the proposed method only requires knowledge of the direct arrival time from a focusing point to the surface and the reflection response of the medium. ... Read more