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

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