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

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

Inverse problems for abstract evolution equations II: higher order differentiability for viscoelasticity

Abstract. In this follow-up of [Inverse Problems 32 (2016) 085001] we generalize our previous abstract results so that they can be applied to the viscoelastic wave equation which serves as a forward model for full waveform inversion (FWI) in seismic imaging including dispersion and attenuation. FWI is the nonlinear inverse problem of identifying parameter functions of the viscoelastic wave equation from measurements of the reflected wave field. Here we rigorously derive rather explicit analytic expressions for the Fréchet derivative and its adjoint (adjoint state method) of the underlying parameter-to-solution map. These quantities enter crucially Newton-like gradient decent solvers for FWI. Moreover, we provide the second Fréchet derivative and a related adjoint as ingredients to second degree solvers

Adaptive Wavelet Collocation Method for Simulation of Time Dependent Maxwell's Equations

This paper investigates an adaptive wavelet collocation time domain method for the numerical solution of Maxwell's equations. In this method a computational grid is dynamically adapted at each time step by using the wavelet decomposition of the field at that time instant. In the regions where the fields are highly localized, the method assigns more grid points; and in the regions where the fields are sparse, there will be less grid points. On the adapted grid, update schemes with high spatial order and explicit time stepping are formulated. The method has high compression rate, which substantially reduces the computational cost allowing efficient use of computational resources. This adaptive wavelet collocation method is especially suitable for simulation of guided-wave optical devices

On the iterative regularization of non-linear illposed problems in L∞

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in $L^{\infty}$ . This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular $L^p$-spaces with $1 < p < \infty$. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smooth Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded and the regularizer, delivered by a discrepancy principle, converges weakly-$\star$? to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis

Approximate inversion of generalized Radon transforms

Generalized Radon transforms (GRT) serve, for instance, as linear models for seismic imaging in the acoustic regime. They occur when the corresponding inverse problem is linearized about a known background compression wave speed (Born approximation). The resulting GRT is completely determined by this background velocity. In this work, we present an implementation of approximate inversion formulas for this class of GRTs proposed and analyzed in [Inverse Problems, 34 (2018), 014002, 114001], where we restrict ourselves to layered background velocities in 2D. In a series of numerical experiments, we intensively test our implementation, reproducing theoretical predictions. Further, we drive the validity of the linearization to its limits

Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime

Time-domain full waveform inversion (FWI) in the acoustic regime comprises a parameter identification problem for the acoustic wave equation: Pressure waves are initiated by sources, get scattered by the earth’s inner structure, and their reflected parts are picked up by receivers located on the surface. From these reflected wave fields the two parameters, density and sound speed, have to be reconstructed. Mathematically, FWI reduces to the solution of a nonlinear and ill-posed operator equation involving the parameter-to-solution map of the wave equation. Newton-like iterative regularization schemes are well suited and well analyzed to tackle this inverse problem. Their convergence results are often based on an assumption about the nonlinear map known as tangential cone condition. In this paper we verify this assumption for a semi-discrete version of FWI where the searched-for parameters are restricted to a finite dimensional space. As a byproduct we establish that the semi-discrete seismic inverse problem is Lipschitz stable, in particular, it is conditionally well-posed