By Anthony J. Devaney
ISBN-10: 052111974X
ISBN-13: 9780521119740
Inverse difficulties are of curiosity and value throughout many branches of physics, arithmetic, engineering and scientific imaging. during this textual content, the principles of imaging and wavefield inversion are provided in a transparent and systematic means. the required conception is progressively built in the course of the booklet, progressing from uncomplicated wave equation dependent types to vector wave versions. by way of combining thought with quite a few MATLAB established examples, the writer promotes a whole realizing of the fabric and establishes a foundation for actual international functions. Key issues of debate comprise the derivation of strategies to the inhomogeneous and homogeneous Helmholtz equations utilizing eco-friendly functionality concepts the propagation and scattering of waves in homogeneous and inhomogeneous backgrounds and the concept that of box time reversal. Bridging the distance among arithmetic and physics, this multidisciplinary e-book will attract graduate scholars and researchers alike.
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Extra info for Mathematical Foundations of Imaging, Tomography and Wavefield Inversion
Sample text
36b) and either one can, in principle, be computed from the other.
35b) where R = (x − a cos φ )2 + (y − a sin φ ) + z2 and kn = ωn /c, with c being the velocity of the background medium. 2 Representation of the radiated field in terms of boundary values via the Kirchhoff–Helmholtz theorem The solution to the radiation problem obtained in the previous section represents the radiated field directly in terms of the source q(r, t). If we now take τ to be a finite region that contains the source region τ0 and has closed boundary ∂τ and restrict our attention to space points lying outside τ , it is possible to represent the radiated field in terms of the field and its normal derivative evaluated on the closed boundary surface ∂τ .
Which translates to k > 0. The poles in the integrand of Eq. 26) occur at K = ±k. p. p. p. p. if z < 0, the integration contours can be closed for both cases and the integral can be computed using residue calculus. 27) e . 2k We can directly verify that the Green function defined in Eq. 27) satisfies the defining equation Eq. 25) by direct differentiation. In particular, we have that G+ (z, ω) = − 1 ∂ G+ (z, ω) = eik|z| Sgn(z); ∂z 2 ∂ ∂2 ik 1 G+ (z, ω) = eik|z| Sgn2 (z) + eik|z| Sgn(z), 2 2 ∂z ∂z2 where Sgn(z) = 1 if z > 0 −1 if z < 0 is the sign function.