Inexact Augmented Lagrangian Method-Based Full-waveform Inversion with Randomized Singular Value Decomposition

Full Waveform Inversion (FWI) is a modeling algorithm used for seismic data processing and subsurface structure inversion. Theoretically, the main advantage of FWI is its ability to obtain useful subsurface structure information, such as velocity and density, from complex seismic data through inversion simulation. However, under complex conditions, FWI is difficult to achieve high-resolution imaging results, and most of the cases are due to random noise, initial model, or inversion parameters and so on. Therefore, we consider an effective image processing and dimension reduction tool, randomized singular value decomposition (rSVD) - weighted truncated nuclear norm regularization (WTNNR), for embedding FWI to achieve high-resolution imaging results. This algorithm obtains a truncated matrix approximating the original matrix by reducing the rank of the velocity increment matrix, thus achieving the truncation of noisy data, with the truncation range controlled by WTNNR. Subsequently, we employ an inexact augmented Lagrangian method (iALM) algorithm in the optimization to compress the solution space range, thus relaxing the dependence of FWI and rSVD-WTNNR on the initial model and accelerating the convergence rate of the objective function. We tested on two sets of synthetic data, and the results show that compared with traditional FWI, our method can more effectively suppress the impact of random noise, thus obtaining higher resolution and more accurate subsurface model information. Meanwhile, due to the introduction of iALM, our method also significantly improves the convergence rate. This work indicates that the combination of rSVD-WTNNR and FWI is an effective imaging strategy which can help to solve the challenges faced by traditional FWI.Comment: 55 Pages, 21 Figure

An all-at-once approach to full wavefrom seismic inversion in the viscoelastic regime

Full waveform seismic inversion (FWI) in the viscoelastic regime entails the task of identifying parameters in the viscoelastic wave equation from partial waveform measurements. Traditionally, one frames this nonlinear problem as an operator equation for the parameter-to-state map. Alternatively, in an all-at-once approach one augments the nonlinear operator by the viscoelastic wave equation as an additional component and considers the states as additional variables. Hence, parameters and states are sought-for simultaneously. In this article, we give a mathematically rigorous all-at-once version of FWI in a functional analytical formulation. Further, the corresponding nonlinear map is shown to be Fréchet differentiable and the adjoint operator of the Fréchet derivative is given in an explicit way suitable for implementation in a Newton-type/gradient-based regularization scheme

Improved full-waveform inversion for seismic data in the presence of noise based on the K-support norm

Full-waveform inversion (FWI) is known as a seismic data processing method that achieves high-resolution imaging. In the inversion part of the method that brings high resolution in finding a convergence point in the model space, a local numerical optimization algorithm minimizes the objective function based on the norm using the least-square form. Since the norm is sensitive to outliers and noise, the method may often lead to inaccurate imaging results. Thus, a new regulation form with a more practical relaxation form is proposed to solve the overfitting drawback caused by the use of the norm,, namely the K-support norm, which has the form of more reasonable and tighter constraints. In contrast to the least-square method that minimizes the norm, our K-support constraints combine the and the norms. Then, a quadratic penalty method is adopted to linearize the non-linear problem to lighten the computational load. This paper introduces the concept of the K-support norm and integrates this scheme with the quadratic penalty problem to improve the convergence and robustness against background noise. In the numerical example, two synthetic models are tested to clarify the effectiveness of the K-support norm by comparison to the conventional norm with noisy data set. Experimental results indicate that the modified FWI based on the new regularization form effectively improves inversion accuracy and stability, which significantly enhances the lateral resolution of depth inversion even with data with a low signal-to-noise ratio (SNR).Comment: 54 pages, 21 figure

An all-at-once approach to full waveform inversion in the viscoelastic regime

Full waveform seismic inversion (FWI) in the viscoelastic regime entails the task of identifying parameters in the viscoelastic wave equation from partial waveform measurements. Traditionally, one frames this nonlinear problem as an operator equation for the parameter‐to‐state map. Alternatively, in an all‐at‐once approach, one augments the nonlinear operator by the viscoelastic wave equation as an additional component and considers the states as additional variables. Hence, parameters and states are sought for simultaneously. In this article, we give a mathematically rigorous all‐at‐once version of FWI in a functional analytical formulation. Further, the corresponding nonlinear map is shown to be Fréchet differentiable, and the adjoint operator of the Fréchet derivative is given in an explicit way suitable for implementation in a Newton‐type/gradient‐based regularization scheme

Towards ultrasound travel time tomography for quantifying human limb geometry and material properties

Sound speed inversions made using simulated time of flight data from a numerical limb-mimicking phantom comprised of soft tissue and a bone inclusion demonstrate that wave front tracking forward modeling combined with 1 regularization could lead to accurate estimates of bone sound-speed. Ultrasonic tomographic imaging of limbs has the potential to impact prosthetic socket fitting, as well as detect and track muscular dystrophy diseases, osteoporosis and bone fractures at low cost and without radiation exposure. Research in ultrasound tomography of bones has increased in the last 10 years, however, methods delivering clinically useful sound-speed inversions are lacking. Inversions for the sound-speed of the numerical phantoms using 1 and 2 regularizations are compared using wave front forward models. The simulations are based on a custom-made cylindrically-scanning tomographic medical ultrasound system (0.5 – 5 MHz) consisting of two acoustic transducers capable of collecting pulse echo and travel time measurements over the entire 360° aperture. Keywords: Ultrasound tomography, bone, migration, reverse time migratio

Seismic Waves

The importance of seismic wave research lies not only in our ability to understand and predict earthquakes and tsunamis, it also reveals information on the Earth's composition and features in much the same way as it led to the discovery of Mohorovicic's discontinuity. As our theoretical understanding of the physics behind seismic waves has grown, physical and numerical modeling have greatly advanced and now augment applied seismology for better prediction and engineering practices. This has led to some novel applications such as using artificially-induced shocks for exploration of the Earth's subsurface and seismic stimulation for increasing the productivity of oil wells. This book demonstrates the latest techniques and advances in seismic wave analysis from theoretical approach, data acquisition and interpretation, to analyses and numerical simulations, as well as research applications. A review process was conducted in cooperation with sincere support by Drs. Hiroshi Takenaka, Yoshio Murai, Jun Matsushima, and Genti Toyokuni

Data-scalable Hessian preconditioning for distributed parameter PDE-constrained inverse problems

Hessian preconditioners are the key to efficient numerical solution of large-scale distributed parameter PDE-constrained inverse problems with highly informative data. Such inverse problems arise in many applications, yet solving them remains computationally costly. With existing methods, the computational cost depends on spectral properties of the Hessian which worsen as more informative data are used to reconstruct the unknown parameter field. The best case scenario from a scientific standpoint (lots of high-quality data) is therefore the worst case scenario from a computational standpoint (large computational cost). In this dissertation, we argue that the best way to overcome this predicament is to build data-scalable Hessian/KKT preconditioners---preconditioners that perform well even if the data are highly informative about the parameter. We present a novel data-scalable KKT preconditioner for a diffusion inverse problem, a novel data-scalable Hessian preconditioner for an advection inverse problem, and a novel data-scalable domain decomposition preconditioner for an auxiliary operator that arises in connection with KKT preconditioning for a wave inverse problem. Our novel preconditioners outperform existing preconditioners in all three cases: they are robust to large numbers of observations in the diffusion inverse problem, large Peclet numbers in the advection inverse problem, and high wave frequencies in the wave inverse problem.Computational Science, Engineering, and Mathematic

Entropy in Image Analysis III

Image analysis can be applied to rich and assorted scenarios; therefore, the aim of this recent research field is not only to mimic the human vision system. Image analysis is the main methods that computers are using today, and there is body of knowledge that they will be able to manage in a totally unsupervised manner in future, thanks to their artificial intelligence. The articles published in the book clearly show such a future