12 research outputs found
Relaxed regularization for linear inverse problems
We consider regularized least-squares problems of the form . Recently, Zheng et al.,
2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3)
that employs a splitting strategy by introducing an auxiliary variable and
solves . By minimizing out the variable we obtain an
equivalent system . 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
as a function of .
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
. 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
Plug-and-Play regularized 3D seismic inversion with 2D pre-trained denoisers
Post-stack seismic inversion is a widely used technique to retrieve
high-resolution acoustic impedance models from migrated seismic data. Its
modelling operator assumes that a migrated seismic data can be generated from
the convolution of a source wavelet and the time derivative of the acoustic
impedance model. Given the band-limited nature of the seismic wavelet, the
convolutional model acts as a filtering operator on the acoustic impedance
model, thereby making the problem of retrieving acoustic impedances from
seismic data ambiguous. In order to compensate for missing frequencies,
post-stack seismic inversion is often regularized, meaning that prior
information about the structure of the subsurface is included in the inversion
process. Recently, the Plug-and-Play methodology has gained wide interest in
the inverse problem community as a new form of implicit regularization, often
outperforming state-of-the-art regularization. Plug-and-Play can be applied to
any proximal algorithm by simply replacing the proximal operator of the
regularizer with any denoiser of choice. We propose to use Plug-and-Play
regularization with a 2D pre-trained, deep denoiser for 2D post-stack seismic
inversion. Additionally, we show that a generalization of Plug-and-Play, called
Multi-Agent Consensus Equilibrium, can be adopted to solve 3D post-stack
inversion whilst leveraging the same 2D pre-trained denoiser used in the 2D
case. More precisely, Multi-Agent Consensus Equilibrium combines the results of
applying such 2D denoiser in the inline, crossline, and time directions in an
optimal manner. We verify the proposed methods on a portion of the SEAM Phase 1
velocity model and the Sleipner field dataset. 1Comment: 24 pages, 10 figure
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