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The “solutions” of boundary-value problems presented in Section 2.8 are merely formal in that the appropriate Green functions must be found, and finding these Green functions is extremely difficult except in certain special cases such as the Rayleigh-Sommerfeld problem. Moreover, even when the appropriate Green functions can be found they are often expressed in the form of superpositions of elementary solutions of the homogeneous Helmholtz equation that are especially suited for dealing with boundaries of a specific shape. One example of a set of such elementary solutions is the plane waves which arise from applying the method of separation of variables to the homogeneous Helmholtz equation using a Cartesian coordinate system and, as we will see below, form a complete set of basis functions for fitting boundary-value data specified on plane surfaces; e.g., for the RS problems. However, the plane waves have limited utility in solving boundary-value problems involving non-planar boundaries such as spherical boundaries. In that case the method of separation of variables is applied to the Helmholtz equation using a spherical polar coordinate system and the so-called multipole fields arise as a set of elementary solutions that form a basis for fitting boundary-value data specified on spherical boundaries. In this chapter we will briefly review the method of separation of variables for the Helmholtz equation and obtain the resulting eigenfunctions for the important cases of Cartesian, spherical polar and cylindrical coordinate systems.
The formulas Eq. (1.33) of Chapter 1 represent the solution to the radiation problem in a non-dispersive medium governed by the wave equation; i.e., they give the radiated field u+(r, t) in terms of a known source q(r, t). These formulas were generalized to dispersive media in Chapter 2, where the radiation problem was solved directly in the frequency domain for a known source embedded in a uniform dispersive background medium. The inverse source problem (ISP), as its name indicates, is the inverse to the radiation problem, and in this problem one seeks the source q(r, t) from knowledge of its radiated field u+(r, t). The question of what applications require a solution to an inverse source problem naturally arises. There are basically two such applications that consist of (i) imaging (reconstructing) the interior of a volume source from observations of the field radiated by the source and (ii) designing a volume source to act as a multi-dimensional antenna to radiate a prescribed field. In the first application actual field measurements are employed, thereby generating data that are then used to “solve” the ISP and thus “reconstruct” the interior of the source, whereas in the second application desired field data are used to “design” a source that will generate those data. Regarding the ISP, the two applications are essentially identical, differing only in emphasis; in application (i) we have to contend with measurement error and noisy data, whereas in application (ii) we have to contend with inconsistencies between the desired data and the constraints required of the source (antenna).
In the radiation problem treated in Chapters 1 and 2 a “source” q(r, t) in the time domain or Q(r, ω) in the frequency domain radiated a wavefield that satisfied either the inhomogeneous wave equation in the time domain or the inhomogeneous Helmholtz equation in the frequency domain. In either case the solution to the radiation problem was easily obtained in the form of a convolution of the given source function with the causal Green function of the wave or Helmholtz equation. A key point concerning the radiation problem is that the source to the radiated field is assumed to be known (specified) and is assumed to be independent of the field that it radiates. Such sources are sometimes referred to as “primary” sources since the mechanism or process that created them is unknown or, at least, unimportant as regards the field that they radiate.
In this chapter we will also encounter the radiation problem, but with sources that are created by the interaction of a propagating wave incident on a physical obstacle or inhomogeneous region of space. These new types of sources are referred to as “induced” or “secondary” sources and the problem of computing the field that they radiate given the incident wave and a model for the field-obstacle interaction is called the scattering problem. We deal with two classes of scattering problem in this book: (i) scattering from so-called “penetrable” scatterers, where the incident wave penetrates into the interior of the obstacle so that the resulting induced source radiates as a conventional volume source of the type treated in earlier chapters; and (ii) scattering from non-penetrable scatterers, where the interaction of the incident wave with the obstacle occurs only over the object's surface.
Inverse problems are of interest and importance across many branches of physics, mathematics, engineering and medical imaging. In this text, the foundations of imaging and wavefield inversion are presented in a clear and systematic way. The necessary theory is gradually developed throughout the book, progressing from simple wave equation based models to vector wave models. By combining theory with numerous MATLAB based examples, the author promotes a complete understanding of the material and establishes a basis for real world applications. Key topics of discussion include the derivation of solutions to the inhomogeneous and homogeneous Helmholtz equations using Green function techniques; the propagation and scattering of waves in homogeneous and inhomogeneous backgrounds; and the concept of field time reversal. Bridging the gap between mathematics and physics, this multidisciplinary book will appeal to graduate students and researchers alike. Additional resources including MATLAB codes and solutions are available online at www.cambridge.org/9780521119740.
I started this book roughly 20 years ago with the intention of producing a finished product within a year or so. But reality in the form of government research grants and “publish or perish” soon set in and so now, at long last, I have finally finished. The final product has of course changed significantly over these intervening years, both in content and in breadth. My original plan was to put together a six- or seven-chapter treatise on basic “Fourier-based” coherent imaging and diffraction tomography complete with Matlab codes implementing the imaging and inversion algorithms presented in the text. The current book certainly includes this material, but also includes a host of other material such as the chapter on time-reversal imaging and the four chapters on the propagation and scattering of waves in homogeneous and inhomogeneous backgrounds. More importantly, the “Fourier-based” inversion schemes originally used to develop much of coherent imaging and linearized inverse scattering (diffraction tomography) have been replaced by the much more powerful singular value decomposition (SVD). This approach allows virtually all of the linearized inverse problems associated with the wave and Helmholtz equation both in homogeneous and in inhomogeneous backgrounds to be treated in a uniform “turn the crank” manner.
My work on imaging and wavefield inversion began as a graduate student under Professor Emil Wolf at the University of Rochester. Originally I had intended to pursue my Ph.D. in quantum optics, but had my plans changed significantly by an off-hand remark by Professor Wolf during one of our meetings.
The “direct” or “forward” scattering problem was treated in the preceding two chapters, where the goal was the computation of a scattered field given knowledge of the scattering object and the incident wavefield. In the “inverse scattering problem” (ISCP) the goal is the determination of the scattering object given knowledge of the incident wave and the scattered wave over some restricted region of space. In Chapter 6 we treated so-called “penetrable” scatterers, where the incident wave penetrates into the interior of the obstacle, thus creating an “induced volume source” that then radiates as a conventional volume source of the type treated in earlier chapters. In Chapter 7 we treated non-penetrable scatterers, where the interaction of the incident wave with the obstacle occurs only over the object's surface. We also treated certain inverse problems associated with non-penetrable scatterers in that chapter that included inverse diffraction and the ISCP of determining the shape of a Dirichlet or Neumann scatterer from its scattering amplitude. In this chapter we will treat the ISCP for penetrable scatterers. We will also make the simplifying assumption that the scattering object is embedded in a uniform lossless medium. This assumption will be discarded in the next chapter, where we will treat scatterers embedded in non-uniform and dispersive media.
We pointed out in Chapter 5 that the difficulty of the “inverse source problem” (ISP) lies in the fact that the radiated field from which the source is to be determined is known only over space points that lie in some restricted region of space that is outside the support of the (unknown) source.
The Green-function solution to the radiation problem given in Eq. (2.23) of Chapter 2 represents this solution in terms of a superposition of outgoing spherical waves with each spherical wave being weighted by the source amplitude at that point. This solution was derived starting from the fact that the Helmholtz equation is linear and, hence, can be represented as a superposition of elementary solutions to the equation when excited by delta functions; i.e., as a convolution of the source term with a Green function that satisfies the same outgoing-wave condition, namely the Sommerfeld radiation condition (SRC), as is satisfied by the radiated field. Alternative representations of the field can also be obtained by making use of the linearity of the Helmholtz equation and the fact that the radiated field satisfies the homogeneous Helmholtz equation everywhere outside the source region τ0. In particular, as we have seen in the last chapter, it is possible to represent the field in such regions in terms of an expansion of eigenfunctions of the homogeneous Helmholtz equation such as the plane waves or multipole fields. Indeed, in Examples 3.3 and 3.5 of Chapter 3 we expanded outgoing-wave fields such as the radiated field in a plane-wave expansion and a multipole expansion, respectively, with the expansion coefficients (planewave amplitudes and multipole moments) determined directly from boundary values of the field. We continue with this task in this chapter, where we develop plane-wave and multipole expansions for the radiated field directly in terms of the source Q rather than in terms of the boundary value of the radiated field.
In this chapter we turn our attention to scattering from non-penetrable objects, or “surface scattering,” and “diffraction” from planar apertures. As was mentioned in the introduction to the previous chapter, the interaction of an incident wave with a non-penetrable scatterer occurs over the surface of the scattering obstacle and is thus defined by some type of boundary condition over this surface. In a similar vein diffraction of an incident wave from apertures cut into non-penetrable surfaces is also defined by some type of boundary condition over the aperture plus surface and thus can, in a certain sense, be considered to be a type of surface scattering. The formal solution to both types of problems is thus obtained in an identical fashion by converting the problem into a boundary-value problem, which is then easily solved using the theory developed in Chapter 2.
The above prescription for “solving” surface scattering and aperture diffraction problems has one missing ingredient: determination of the boundary values required in the solution of the scattering or diffraction problem. This is the ingredient that distinguishes a scattering or diffraction problem from the purely mathematical boundary-value problem. In this chapter we will restrict our attention to non-penetrable objects over which the total field (incident plus scattered) satisfies homogeneous Dirichlet or Neumann conditions. By invoking this condition it is possible to represent the scattered field in terms of either the value of the normal derivative of the total field (the homogeneous Dirichlet case) or the total field itself (the homogeneous Neumann case) over the scatterer surface.
We return to the problem of computing the field u+(r, t) radiated by a real-valued spaceand time-varying source q(r, t) embedded in an infinite homogeneous medium such as free space. As in Chapter 1 we will assume here that the time-dependent source q(r, t) is compactly supported in the space-time region {S0|r ϵ τ0, ϵ t ϵ [0, T0]}, where τ0 is its spatial volume and [0, T0] the interval of time over which the source is turned on. In the case in which the medium is non-dispersive the radiated wavefield satisfies the inhomogeneous scalar wave equation Eq. (1.1). More generally, if the background medium is dispersive it is necessary to replace the second time derivative in this equation by an integral (convolutional) operator, so that the wave equation is actually an integral-differential equation. In this chapter we will treat the radiation problem in the frequency domain so that this complication is avoided and our results apply both to dispersive and to non-dispersive backgrounds.
In addition to treating the radiation problem we also treat the classical boundary value problem for the scalar wave Helmholtz equation in a (possibly dispersive) uniform background medium. Special attention is devoted to the famous Rayleigh–Sommerfeld boundary-value problem, which consists of computing a radiated field throughout a half-space that is exterior to the source region τ0 from Dirichlet or Neumann conditions prescribed over an infinite bounding plane to the source.
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