Kirchhoff inversion

The corresponding parametric functional for Kirchhoff inversion with a minimum norm stabilizer has the form... 

In the general case of an iterative inversion scheme (15.135), we should apply the adjoint Kirchhoff operator K to the residual field Rn = K(A ) — da computed on each iteration. Each of these applications is equivalent to the standard migration of the residual field. Thus, iterative Kirchhoff inversion can be treated as an iterative migration algorithm,... 

In the section of this chapter describing the Kirchhoff inversion method and general non-linear inversion techniques, we have demonstrated that the calculation of the Kirchhoff adjoint operator (15.142) and the adjoint Frechet derivative operator... 

Kirchhoff s applications of these laws to the spectrum of the sun are well known. A body emitting temperature radiation absorbs all the wave lengths which it emits, and vice versa. A sodium flame, for example, must absorb light of all the wave lengths which are emitted by another sodium flame. (Inversion of the Na-line.) In this way Kirchhoff arrived at his exceedingly fruitful theory of the Fraunhofer lines in the spectrum of the sun. 

A similar result holds true for the case shown in Figure 13-4b. Applying the inverse Fourier transform to both sides of the last expression, we cirrive at the Kirchhoff integral formula for an unbounded domain ... 

For practical realization of this iterative inversion scheme we have to find the explicit expressions for the adjoint Kirchhoff operator K according to formula... 

Another method of practical realization of the Kirchhoff type reverse-time migration is based on the Rayleigh integral formula in the frequency domain (15.200). Applying an inverse Fourier transform to both sides of the Rayleigh formula, we obtain for f = 0... 

M. Kunz, Inverses of perturbed Laplace-Kirchhoff and some distance matrices, MATCH, Commun. Math. Comput. Chem. 32 (1995) 221-234. 

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