Changchun Yang

Optimization and regularization for computational inverse problems and applications

Introduction.- Regularization Theory and Recent Developments.- Nonstandard Regularization and Advanced Optimization Theory and Methods.- Numerical Inversion in Geoscience and Quantitative Remote Sensing.

A review on restoration of seismic wavefields based on regularization and compressive sensing

Restoration of seismic data as an ill-posed inverse problem means to recover the complete wavefields from sub-sampled data. Since seismic data are typically sparse in the curvelet domain, this problem can be solved based on the compressive sensing theory. Meanwhile three major problems are modelling, sampling and solving methods. We first construct l 0 and l 1 minimization models and then develop fast projected gradient methods to solve the restoration problem. For seismic data interpolation/restoration, the regular sub-sampled data will generate coherence aliasing in the frequency domain, while the random sub-sampling cannot control the largest sampling gap. Therefore, we consider a new sampling technique in this article which is based on the controlled piecewise random sub-sampling scheme. Numerical simulations are made and compared with the iterative soft thresholding method and the spectral gradient-projection method. It reveals that the proposed algorithms have the advantages of high precision, robustness and fast calculation.

Regularizing active set method for nonnegatively constrained ill-posed multichannel image restoration problem

In this paper, we consider the nonnegatively constrained multichannel image deblurring problem and propose regularizing active set methods for numerical restoration. For image deblurring problems, it is reasonable to solve a regularizing model with nonnegativity constraints because of the physical meaning of the image. We consider a general regularizing l(p)-l(q) model with nonnegativity constraints. For p and q equaling 2, the model is in a convex quadratic form, therefore, the active set method is proposed since the nonnegativity constraints are imposed naturally. For p and q not equaling 2, we present an active set method with a feasible Newton-conjugate gradient solution technique. Numerical experiments are presented for ill-posed three-channel blurred image restoration problems.

ON TIKHONOV REGULARIZATION AND COMPRESSIVE SENSING FOR SEISMIC SIGNAL PROCESSING

Using compressive sensing and sparse regularization, one can nearly completely reconstruct the input (sparse) signal using limited numbers of observations. At the same time, the reconstruction methods by compressing sensing and optimizing techniques overcome the obstacle of the number of sampling requirement of the Shannon/Nyquist sampling theorem. It is well known that seismic reflection signal may be sparse, sometimes and the number of sampling is insufficient for seismic surveys. So, the seismic signal reconstruction problem is ill-posed. Considering the ill-posed nature and the sparsity of seismic inverse problems, we study reconstruction of the wavefield and the reflection seismic signal by Tikhonov regularization and the compressive sensing. The l0, l1 and l2 regularization models are studied. Relationship between Tikhonov regularization and the compressive sensing is established. In particular, we introduce a general lp - lq (p, q ≥ 0) regularization model, which overcome the limitation on the assumption of convexity of the objective function. Interior point methods and projected gradient methods are studied. To show the potential for application of the regularized compressive sensing method, we perform both synthetic seismic signal and field data compression and restoration simulations using a proposed piecewise random sub-sampling. Numerical performance indicates that regularized compressive sensing is applicable for practical seismic imaging.

Determining finite difference weights for the acoustic wave equation by a new dispersion‐relationship‐preserving method

Numerical simulation of the acoustic wave equation is widely used to theoretically synthesize seismograms and constitutes the basis of reverse-time migration. With finite-difference methods, the discretization of temporal and spatial derivatives in wave equations introduces numerical grid dispersion. To reduce the grid dispersion effect, we propose to satisfy the dispersion relation for a number of uniformly distributed wavenumber points within a wavenumber range with the upper limit determined by the maximum source frequency, the grid spacing and the wave velocity. This new dispersion-relationship-preserving method relatively uniformly reduces the numerical dispersion over a large-frequency range. Dispersion analysis and seismic numerical simulations demonstrate the effectiveness of the proposed method.

Computational Methods for Applied Inverse Problems

This monograph reports recent advances of inversion theory and recent developments with practical applications in frontiers of sciences, especially inverse design and novel computational methods for inverse problems. Readers who do research in applied mathematics, engineering, geophysics, biomedicine, image processing, remote sensing, and environmental science will benefit from the contents since the book incorporates a background of using statistical and non-statistical methods, e.g., regularization and optimization techniques for solving practical inverse problems.

Comparison of numerical dispersion in acoustic finite-difference algorithms

Numerical simulation of the acoustic wave equation is widely used to synthesise seismograms theoretically, and is also the basis of the reverse time migration. With some stability conditions, grid dispersion often exists because of the discretisation of the time and the spatial derivatives in the wave equation. How to suppress the grid dispersion is therefore a key problem for finite-difference approaches. Different methods are proposed to address the problem. The commonly used methods are the high order Taylor expansion methods and the optimised methods. In this paper, we compare the performance of these methods in the space and time–space domains. We demonstrate by dispersion analysis and numerical simulation that a linear method without iteration performs comparably to the optimised methods, but with reduced computational effort. Different methods were developed to suppress the grid dispersion in numerical simulation of the acoustic wave equation. This paper compares the performance of these methods in the space and time–space domains. Dispersion analysis and numerical simulation indicate that a linear method without iteration performs comparably to the optimised methods.

The Goos-Hänchen shift of wide-angle seismic reflection wave

The partial derivative equations of Zoeppritz equations are established and the derivatives of each matrix entry with respect to wave vectors are derived in this paper. By solving the partial derivative equations we obtained the partial derivatives of seismic wave reflection coefficients with respect to wave vectors, and computed the Goos-Hänchen shift for reflected P- and VS-waves. By plotting the curves of Goos-Hänchen shift, we gained some new insight into the lateral shift of seismic reflection wave. The lateral shifts are very large for glancing wave or the wave of the incidence angle near the critical angle, meaning that the seismic wave propagates a long distance along the reflection interface before returning to the first medium. For the reflection waves of incidence angles away from the critical angle, the lateral shift is in the same order of magnitude as the wavelength. The lateral shift varies significantly with different reflection interfaces. For example, the reflected P-wave has a negative shift at the reflection interface between mudstone and sandstone. The reflected VS-wave has a large lateral shift at or near the critical angle. The lateral shift of the reflected VS-wave tends to be zero when the incidence angle approaches 90°. These observations suggest that Goos-Hänchen effect has a great influence on the reflection wave of wide-angles. The correction for the error caused by Goos-Hänchen effect, therefore, should be made before seismic data processing, such as the depth migration and the normal-moveout correction. With the theoretical foundation established in this paper, we can further study the correction of Goos-Hänchen effect for the reflection wave of large incidence angle.

The Coupling Relationship between the West Shandong Rise and the Jiyang Depression, China

This article reports 21 AFT (apatite fission track) data from the West Shandong (山东) rise (WSR) and Jiyang (济阳) depression, and mainly studies their Cenozoic uplifting/subsidence history and the relationship between them. Furthermore, we improve our insights into the Bohai Bay Basin (BBB). Our AFT analysis and AFT T-t modeling indicates that the WSR was uplifted at ca. 65 Ma with apparent uplift rate of 0.019 mm/a; it underwent two relatively rapid uplifting events at 43–33 and 16–0 Ma with rates of 0.097 and 0.052 mm/a, respectively. Meanwhile, the Jiyang depression subsided at rate of 0.032 mm/a at 52–43 Ma, and the rate increased to 0.13 mm/a at ca. 42–33 Ma; finally the subsidence rate increased to 0.053 mm/a in 16–0 Ma. They all underwent a uplift in time of 23–16 Ma with rate of 0.04–0.07 mm/a. A careful comparison shows that the Cenozoic uplifting of the WSR coupled well with the subsidence of the Jiyang depression. Our research also suggests that the uplift-basin coupling events are part of the couplings between the Bohai Bay Basin and its peripheral mountains. This intraplate mountain-basin coupling is a reflection of global tectonic events.

Inverse Problems, Optimization and Regularization: A Multi-Disciplinary Subject

Inverse problems, optimization, regularization and scientific computing as a multi-disciplinary subject are introduced in this introductory chapter.