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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 integmis are computzd by Thor method of stationary phase. The output is asymptotically a scaled singulnrfunction 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 then 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. INTRODUCTION This paper is motivated by the brilliant paper by Beylkin (1985). Beylkin presented a theory for asymptotic inversion of observations for the constant-density acoustic wave equation. The method is based on the Born approximation for the forward problem. It allows for a completely general background sound speed in the inverse problem, as well as an assortment of possible source-receiver configurations broad enough to accommodate most of the cases of interest in seismic exploration and other applications. For example, the method applies to zero-offset data; common-source (or single-source), multireceiver array data (or the reverse); or fixed-offset data. The...

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.

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This paper collects certain results concerning wave propagation in two‐and‐one‐half dimensions, i.e., three‐dimensional (3‐D) wave propagation in a medium that has variations in two dimensions only. The results of interest are for sources and receivers in the plane determined by the two directions of parameter variation. The objective of this work is to reduce the analysis of the in‐plane propagation to 2‐D analysis while retaining–at least asymptotically–the proper 3‐D geometrical spreading. We do this for the free space Green's function and for the Kirchhoff approximate upward scattered field from a single reflector. In both cases the derivation is carried out under the assumption of a background velocity c(x, z) with the special cases c = c0 and c = c(z).