Relaxed regularization for linear inverse problems

We consider regularized least-squares problems of the form $\min_{x} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \mathcal{R}(Lx)$. Recently, Zheng et al., 2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable $y$ and solves $\min_{x,y} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \frac{\kappa}{2}\Vert Lx - y\Vert_2^2 + \mathcal{R}(x)$. By minimizing out the variable $x$ we obtain an equivalent system $\min_{y} \frac{1}{2} \Vert F_{\kappa}y - g_{\kappa}\Vert_2^2+\mathcal{R}(y)$. In our work we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of $F_{\kappa}$ as a function of $\kappa$. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem and we quantify the error incurred by relaxation in terms of $\kappa$. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach.Comment: 25 pages, 14 figures, submitted to SIAM Journal for Scientific Computing special issue Sixteenth Copper Mountain Conference on Iterative Method

Seeing through the CO2 plume: joint inversion-segmentation of the Sleipner 4D Seismic Dataset

4D seismic inversion is the leading method to quantitatively monitor fluid flow dynamics in the subsurface, with applications ranging from enhanced oil recovery to subsurface CO2 storage. The process of inverting seismic data for reservoir properties is, however, a notoriously ill-posed inverse problem due to the band-limited and noisy nature of seismic data. This comes with additional challenges for 4D applications, given inaccuracies in the repeatability of the time-lapse acquisition surveys. Consequently, adding prior information to the inversion process in the form of properly crafted regularization terms is essential to obtain geologically meaningful subsurface models. Motivated by recent advances in the field of convex optimization, we propose a joint inversion-segmentation algorithm for 4D seismic inversion, which integrates Total-Variation and segmentation priors as a way to counteract the missing frequencies and noise present in 4D seismic data. The proposed inversion framework is applied to a pair of surveys from the open Sleipner 4D Seismic Dataset. Our method presents three main advantages over state-of-the-art least-squares inversion methods: 1. it produces high-resolution baseline and monitor acoustic models, 2. by leveraging similarities between multiple data, it mitigates the non-repeatable noise and better highlights the real time-lapse changes, and 3. it provides a volumetric classification of the acoustic impedance 4D difference model (time-lapse changes) based on user-defined classes. Such advantages may enable more robust stratigraphic and quantitative 4D seismic interpretation and provide more accurate inputs for dynamic reservoir simulations. Alongside our novel inversion method, in this work, we introduce a streamlined data pre-processing sequence for the 4D Sleipner post-stack seismic dataset, which includes time-shift estimation and well-to-seismic tie.Comment: This paper proposes a novel algorithm to jointly regularize a 4D seismic inversion problem and segment the 4D difference volume into percentages of acoustic impedance changes. We validate our algorithm with the 4D Sleipner seismic dataset. Furthermore, this paper comprehensively explains the data preparation workflow for 4D seismic inversio

Laterally constrained low-rank seismic data completion via cyclic-shear transform

A crucial step in seismic data processing consists in reconstructing the wavefields at spatial locations where faulty or absent sources and/or receivers result in missing data. Several developments in seismic acquisition and interpolation strive to restore signals fragmented by sampling limitations; still, seismic data frequently remain poorly sampled in the source, receiver, or both coordinates. An intrinsic limitation of real-life dense acquisition systems, which are often exceedingly expensive, is that they remain unable to circumvent various physical and environmental obstacles, ultimately hindering a proper recording scheme. In many situations, when the preferred reconstruction method fails to render the actual continuous signals, subsequent imaging studies are negatively affected by sampling artefacts. A recent alternative builds on low-rank completion techniques to deliver superior restoration results on seismic data, paving the way for data kernel compression that can potentially unlock multiple modern processing methods so far prohibited in 3D field scenarios. In this work, we propose a novel transform domain revealing the low-rank character of seismic data that prevents the inherent matrix enlargement introduced when the data are sorted in the midpoint-offset domain and develop a robust extension of the current matrix completion framework to account for lateral physical constraints that ensure a degree of proximity similarity among neighbouring points. Our strategy successfully interpolates missing sources and receivers simultaneously in synthetic and field data

Seismic wavefield redatuming with regularized multi-dimensional deconvolution

In seismic imaging the aim is to obtain an image of the subsurface using reflection data. The reflection data are generated using sound waves and the sources and receivers are placed at the surface. The target zone, for example an oil or gas reservoir, lies relatively deep in the subsurface below several layers. The area above the target zone is called the overburden. This overburden will have an imprint on the image. Wavefield redatuming is an approach that removes the imprint of the overburden on the image by creating so-called virtual sources and receivers above the target zone. The virtual sources are obtained by determining the impulse response, or Green's function, in the subsurface. The impulse response is obtained by deconvolving all up- and downgoing wavefields at the desired location. In this paper, we pose this deconvolution problem as a constrained least-squares problem. We describe the constraints that are involved in the deconvolution and show that they are associated with orthogonal projection operators. We show different optimization strategies to solve the constrained least-squares problem and provide an explicit relation between them, showing that they are in a sense equivalent. We show that the constrained least-squares problem remains ill-posed and that additional regularizati

Comparing RSVD and Krylov methods for linear inverse problems

In this work we address regularization parameter estimation for ill-posed linear inverse problems with an penalty. Regularization parameter selection is of utmost importance for all of inverse problems and estimating it generally relies on the experience of the practitioner. For regularization with an penalty there exist a lot of parameter selection methods that exploit the fact that the solution and the residual can be written in explicit form. Parameter selection methods are functionals that depend on the regularization parameter where the minimizer is the desired regularization parameter that should lead to a good solution. Evaluation of these parameter selection methods still requires solving the inverse problem multiple times. Efficient evaluation of the parameter selection methods can be done through model order reduction. Two popular model order reduction techniques are Lanczos based methods (a Krylov subspace method) and the Randomized Singular Value Decomposition (RSVD). In this work we compare the two approaches. We derive error bounds for the parameter selection methods using the RSVD. We compare the performance of the Lanczos process versus the performance of RSVD for efficient parameter selection. The RSVD algorithm we use i

Relaxed regularization for linear inverse problems

We consider regularized least-squares problems of the form min {equation presented}. Recently, Zheng et al. [IEEE Access, 7 (2019), pp. 1404-1423], proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable y and solves min {equation presented}. By minimizing out the variable x, we obtain an equivalent optimization problem min {equation presented}. In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of Fκ as a function of κ. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem, and we quantify the error incurred by relaxation in terms of κ. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach

Estimating the regularization parameter efficiently

We consider linear inverse problems with a two norm regularization, called Tikhonov regularization. When using regularization to solve an inverse problem, a regularization parameter is introduced. The regularization parameter heavily controls the quality of the regularized solution. We show various methods to estimate the regularization parameter known from literature that arise in various applications. Furthermore, we will describe how to efficiently evaluate them. We show results on a 2D seismic travel time tomography problem

Relaxed regularization for linear inverse problems

We consider regularized least-squares problems of the form min x 1/2 kAx − bk22+ R(Lx). Recently, Zheng et al. [45] proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable y and solves minx,y 1 2 kAx − bk22 + k/2 2 kLx − yk22 + R(x). By minimizing out the variable x, we obtain an equivalent optimization problem miny 1 2 kFy − gk22 + R(y). In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of F as a function of κ. Furthermore, we relate the Pareto curve of the original problem to the relaxed proble

A hybrid approach to seismic deblending: when physics meets self-supervision

To limit the time, cost, and environmental impact associated with the acquisition of seismic data, in recent decades considerable effort has been put into so-called simultaneous shooting acquisitions, where seismic sources are fired at short time intervals between each other. As a consequence, waves originating from consecutive shots are entangled within the seismic recordings, yielding so-called blended data. For processing and imaging purposes, the data generated by each individual shot must be retrieved. This process, called deblending, is achieved by solving an inverse problem which is heavily underdetermined. Conventional approaches rely on transformations that render the blending noise into burst-like noise, whilst preserving the signal of interest. Compressed sensing type regularization is then applied, where sparsity in some domain is assumed for the signal of interest. The domain of choice depends on the geometry of the acquisition and the properties of seismic data within the chosen domain. In this work, we introduce a new concept that consists of embedding a self-supervised denoising network into the Plug-and-Play (PnP) framework. A novel network is introduced whose design extends the blind-spot network architecture of [28 ] for partially coherent noise (i.e., correlated in time). The network is then trained directly on the noisy input data at each step of the PnP algorithm. By leveraging both the underlying physics of the problem and the great denoising capabilities of our blind-spot network, the proposed algorithm is shown to outperform an industry-standard method whilst being comparable in terms of computational cost. Moreover, being independent on the acquisition geometry, our method can be easily applied to both marine and land data without any significant modification.Comment: 21 pages, 15 figure

Relaxed regularization for linear inverse problems

We consider regularized least-squares problems of the form min {equation presented}. Recently, Zheng et al. [IEEE Access, 7 (2019), pp. 1404-1423], proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable y and solves min {equation presented}. By minimizing out the variable x, we obtain an equivalent optimization problem min {equation presented}. In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of Fκ as a function of κ. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem, and we quantify the error incurred by relaxation in terms of κ. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach