Search for: All records

This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important partial differential equations (PDEs) in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under theL^pnorms, we study these quantities under the SobolevH^snorm, which associates with a general class ofL²-based function spaces. TheH^snorm has a natural spectral bias based on its definition in the Fourier domain: the cases < 0 biases towards the lower frequencies, while the cases > 0 biases towards the higher frequencies. We compare the stability estimates using differentH^snorms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures. We also provide numerical inversion examples to demonstrate the differences in the inverse scattering radius of convergence under different metric spaces.

 

Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics. 

State-of-the-art seismic imaging techniques treat inversion tasks such as full-waveform inversion (FWI) and least-squares reverse time migration (LSRTM) as partial differential equation-constrained optimization problems. Due to the large-scale nature, gradient-based optimization algorithms are preferred in practice to update the model iteratively. Higher-order methods converge in fewer iterations but often require higher computational costs, more line-search steps, and bigger memory storage. A balance among these aspects has to be considered. We have conducted an evaluation using Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate the steepest-descent algorithm, which we innovatively treat as a fixed-point iteration. Independent of the unknown parameter dimensionality, the computational cost of implementing the method can be reduced to an extremely low dimensional least-squares problem. The cost can be further reduced by a low-rank update. We determine the theoretical connections and the differences between AA and other well-known optimization methods such as L-BFGS and the restarted generalized minimal residual method and compare their computational cost and memory requirements. Numerical examples of FWI and LSRTM applied to the Marmousi benchmark demonstrate the acceleration effects of AA. Compared with the steepest-descent method, AA can achieve faster convergence and can provide competitive results with some quasi-Newton methods, making it an attractive optimization strategy for seismic inversion. 

Full waveform inversion (FWI) and least-squares reverse time migration (LSRTM) are popular imaging techniques that can be solved as PDE-constrained optimization problems. Due to the large-scale nature, gradient- and Hessian-based optimization algorithms are preferred in practice to find the optimizer iteratively. However, a balance between the evaluation cost and the rate of convergence needs to be considered. We propose the use of Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate a gradient descent method. We show that AA can achieve fast convergence that provides competitive results with some quasi-Newton methods. Independent of the dimensionality of the unknown parameters, the computational cost of implementing the method can be reduced to an extremely lowdimensional least-squares problem, which makes AA an attractive method for seismic inversion. 

Seismic full-waveform inversion aims to reconstruct subsurface medium parameters from recorded seismic data. It is solved as a constrained optimization problem in the deterministic approach. Many different objective functions have been proposed to tackle the nonconvexity that originated from the cycle-skipping issues. The analogy between objective functions in the deterministic inversion and likelihood functions in Bayesian inversion motivates us to analyze the noise model each objective function accounts for under the Bayesian inference setting. We also show the existence and wellposedness of their corresponding posterior measures. In particular, the theorem shows that theWasserstein-type likelihood offers better stability with respect to the noise in the recorded data. Together with an application of the level-set prior, we demonstrate by numerical examples the successful reconstruction from Bayesian full-waveform inversion under the proper choices of the likelihood function and the prior distribution.