Inverse Problems TOPICAL REVIEW Related content - A comparison of seismic velocity inversion The seismic reflection inverse problem methods for layered acoustics - A mathematical framework for inverse To cite this article: W W Symes 2009 Inverse Problems 25 123008 wave problems in heterogeneous media - A method for inverse scattering based on the generalized Bremmer coupling series View the article online for updates and enhancements. Recent citations - On quasi-seismic wave propagation in highly anisotropic triclinic layer between distinct semi-infinite triclinic geomedia Pato Kumari and Neha - Reduced Order Model Approach to Inverse Scattering Liliana Borcea et al - Seismic wavefield redatuming with regularized multi-dimensional deconvolution Nick Luiken and Tristan van Leeuwen This content was downloaded from IP address 189.46.171.20 on 21/10/2020 at 04:11 IOP PUBLISHING INVERSE PROBLEMS Inverse Problems 25 (2009) 123008 (39pp) doi:10.1088/0266-5611/25/12/123008 TOPICAL REVIEW The seismic reflection inverse problem W W Symes Computational and Applied Mathematics, Rice University, MS 134, Rice University, Houston, TX 77005, USA E-mail: [email protected] Received 3 July 2009, in final form 2 September 2009 Published 1 December 2009 Online at stacks.iop.org/IP/25/123008 Abstract The seismic reflection method seeks to extract maps of the Earth’s sedimentary crust from transient near-surface recording of echoes, stimulated by explosions or other controlled sound sources positioned near the surface. Reasonably accurate models of seismic energy propagation take the form of hyperbolic systems of partial differential equations, in which the coefficients represent the spatial distribution of various mechanical characteristics of rock (density, stiffness, etc). Thus the fundamental problem of reflection seismology is an inverse problem in partial differential equations: to find the coefficients (or at least some of their properties) of a linear hyperbolic system, given the values of a family of solutions in some part of their domains. The exploration geophysics community has developed various methods for estimating the Earth’s structure from seismic data and is also well aware of the inverse point of view. This article reviews mathematical developments in this subject over the last 25 years, to show how the mathematics has both illuminated innovations of practitioners and led to new directions in practice. Two themes naturally emerge: the importance of single scattering dominance and compensation for spectral incompleteness by spatial redundancy. 1. Introduction Reflection seismology is the main tool used by the oil and gas industry to map petroleum deposits in the Earth’s upper crust. Environmental and civil engineers also use variants of the technique to locate bedrock, aquifers and other near-surface features, and academic geophysicists have extended it to a tool for imaging the lower crust and mantle. Seismic wave propagation, the physical phenomenon underlying the reflection method as well as other types of seismology, is modeled with reasonable accuracy as small-amplitude displacement of a continuum, using various specializations and generalizations of linear elastodynamics. In these models, the various mechanical properties of rock regulating the wave propagation phenomenon appear as spatially varying coefficients in a system of time- dependent hyperbolic linear partial differential equations (‘PDEs’). (Somewhat confusingly, 0266-5611/09/123008+39$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1 Inverse Problems 25 (2009) 123008 Topical Review the literature on this subject also uses the word ‘model’ to connote the collection of coefficients characterizing an instance of a continuum mechanical model. I shall also use the word in this sense.) Thus the main goal of reflection seismology could be construed as the solution of inverse problems for certain PDEs. The data of the reflection seismology inverse problem are samples of the solutions of these PDEs, in the form of time series of pressure or related quantities collected at points corresponding to locations of seismic sensors (the common name for these is ‘receivers’). The solution of the inverse problem is the model (collection of coefficient functions), and the implicit relation between data and solution is the system of PDEs. It should also be noted that seismic surveys are multiexperiments: that is, they consist of a number of more or less identical individual experiments (commonly called ‘shots’), from each of which part of the data is collected. In modeling terms, the data are samples taken from solutions of a family of linear PDEs, sharing the same coefficients but with different (and, to some extent, known) right-hand sides. Geophysicists have developed a variety of methods for the approximate solution of this inverse problem. In many cases, these methods were not originally construed as solutions of inverse problems for PDEs, but as processes which convert seismic data to images of the Earth’s interior. The purpose of this review is to overview the mathematical understanding of these seismic imaging methods that has developed almost entirely over the last 25 years. This understanding has both illuminated the rationale and performance of methods used in practice and suggested improvements and alternatives to widely used processes as well as algorithms to extract previously inaccessible information from the data. Amongst other things, the mathematical meaning of the word ‘imaging’ has become much clearer, along with the relation between imaging and inversion. New approaches to solution of inverse problems for hyperbolic PDEs have also emerged from this study. Two central themes pervade this subject. The first is the importance of single scattering, which in mathematical terms is more or less equivalent to linearization of the equations of motion with respect to the model (coefficients), at a smooth model. That single scattering is apparent at all is somewhat surprising, since rocks are typically highly heterogeneous, exhibiting spatial structure at all scales from micrometers to kilometers. Given an appropriate relation between wavelengths, travel distances and correlation lengths of material structures, elastodynamics can produces diffuse fields (Weaver and Lobkis 2006), in which the singly scattered waves are submerged in a sea of multiple scattering. A similar phenomenon occurs in optical tomography of biological tissue, in which ballistic photons typically play an insignificant role (Arridge 1999). The onset of elastodynamic diffusion may in fact limit the scope of the seismic reflection method (Delprat-Jannaud and Lailly 2005). On the other hand, frequencies, length scales, times and distances appear to conspire to produce at least discernible, if not dominant, single scattering in much reflection data from the upper crust. This is the fact upon which the state of the art in reflection seismic imaging is founded. The linearization of the equations of motion with respect to the coefficients results in small error when the background or reference model is smooth, and the perturbation is oscillatory. (The words ‘smooth’ and ‘oscillatory’ are vague—a precise sense for this decomposition is discussed below.) Linearization defines a linear map from coefficient perturbations to data perturbations (sampled solution perturbations). This linear map has been studied extensively in the last 25 years, and the story that has emerged is one of the two main topics of this review. The main results follow from its oscillatory integral representation: it is a so-called Fourier integral operator, a class of oscillatory integral operators developed by Hormander¨ and others to represent the solutions of non-elliptic PDEs (Hormander¨ 1971, 1983). These operators have singularity mapping properties which facilitate an explanation of the relation between imaging and inversion, and indeed motivate a definition of imaging. Critical geological features— 2 Inverse Problems 25 (2009) 123008 Topical Review transitions over very short distances from one type of rock to another with different mechanical characteristics—amount to singular (nonsmooth) or approximately singular features in the coefficient functions. If one defines an image of an Earth structure model as any function which has the same singularities (precise definition below) as an algebraic combination of the coefficient functions of a model, then the mapping properties of the linearized operator explain why solution of the linearized inverse problem results in an image. Also, the imaging processes developed in the seismic literature are approximations in various senses of the linearized inverse, retaining its singularity mapping properties. This picture developed starting in the 1980s with the work of Beylkin (1985), Rakesh (1988), Beylkin and Burridge (1990) and others. The relation between widely used imaging (‘migration’) methods and linearized inversion was already implicit in the pathbreaking work of Cohen and Bleistein (1977, 1979) and was further developed by Lailly (1983, 1984) and many others. Bleistein (1999) provides an overview of this history; I give another below. The second pervasive theme is the compensation for data incompleteness in one sense (spectral) by data redundancy in another sense (spatial). For excellent physical reasons, seismic time series are spectrally incomplete—Fourier components at both very low and very high frequencies are below the noise level. The lack of very high frequencies means that singularities cannot actually