Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

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An overview of full-waveform inversion in exploration geophysics

SUMMARY
Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Frechet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Frechet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint-state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved.

Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

Fractional Calculus: Integral and Differential Equations of Fractional Order

Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

https://iopscience.iop.org/article/10.1088/0034-4885/35/1/306/pdf

Semiclassical approximations in wave mechanics

Asymptotic Expansions of Integrals

Preface.- Multi-Dimensional Seismic Inversion.- The One- Dimensional Inverse Problem.- Inversion in Higher Dimensions.- Large-Wavenumber Fourier Imaging.- Inversion in Heterogeneous Media.- Two-and-One Half Dimensional Inversion.- The General Theory of Data Mapping.- Distribution Theory.- The Fourier Transform of Causal Functions.- Dimensional vs. Dimensionless Variables.- An Example of Ill-Posedness.-An Elementary Introduction to Ray Theory.- Author Index.- Subject Index.

Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

In this paper, I present a modification of the Beylkin inversion operator. This modification accounts for the band-limited nature of the data and makes the role of discontinuities in the sound speed more precise. The inversion presented here partially dispenses with the small-parameter constraint of the Born approximation. This is shown by applying the proposed inversion operator to upward scattered data represented by the Kirchhoff approximation, using the angularly dependent geometrical-optics reflection coefficient. A fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these Kirchhoff-approximate data for the forward problem. The inversion of these data involves integration over the source-receiver surface, the reflecting surface, and frequency. The spatial integrals are computed by the method of stationary phase. The output is asymptotically a scaled singular function of the reflecting surface. The singular function of a surface is a Dirac delta function whose support is on the surface. Thus, knowledge of the singular functions is equivalent to mathematical imaging of the reflector. The scale factor multiplying the singular function is proportional to the geometrical-optics reflection coefficient. In addition to its dependence on the variations in sound speed, this reflection coefficient depends on an opening angle between rays from a source and receiver pair to the reflector. I show how to determine this unknown angle. With the angle determined, the reflection coefficient contains only the sound speed below the reflector as an unknown, and it can be determined. A recursive application of the inversion formalism is possible. That is, starting from the upper surface, each time a major reflector is imaged, the background sound speed is updated to account for the new information and data are processed deeper into the section until a new major reflector is imaged. Hence, the present inversion formalism lends itself to this type of recursive implementation. The inversion proposed here takes the form of a Kirchhoff migration of filtered data traces, with the space-domain amplitude and frequency-domain filter deduced from the inversion theory. Thus, one could view this type of inversion and parameter estimation as a Kirchhoff migration with careful attention to amplitude.

On the imaging of reflectors in the earth

First-Order Partial Differential Equations. The Dirac Delta Function, Fourier Transforms, and Asymptotics. Second-Order Partial Differential Equations. The Wave Equation in One Space Dimension. The Wave Equation in Two and Three Dimensions. The Helmholtz Equation and Other Elliptic Equations. More on Asymptotic Techniques for Direct Scattering Problems. Inverse Methods for Reflector Imaging. Each chapter includes references. Index.

Mathematical Methods for Wave Phenomena

A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Norman Bleistein

Papers

Asymptotic Expansions of Integrals

Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion

On the imaging of reflectors in the earth

Mathematical Methods for Wave Phenomena

Nonuniqueness in the inverse source problem in acoustics and electromagnetics