The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed‐form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green’s function for the background medium. This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one‐way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image highlights rapid changes in the scattering potential (interfaces), whereas the reconstructed image can, under ideal conditions, be a perfect reconstruction of the scattering potential. “Ideal” conditions correspond to (1) weak scattering about a smoothly varying background medium, (2) a receiver array with full angular aperture, and (3) data of infinite bandwidth. Images obtained from a multioffset vertical seismic profile (VSP) illustrate some of the practical differences between the two imaging algorithms. The reconstructed image shows a much clearer picture of the target (a reef structure), in part because the one‐way imaging operator eliminates artifacts caused by the limited aperture of the receiver array.

Downgoing waves in multicomponent VSP experiments are used to obtain seismic P- and S-wave velocities as a function of depth and angle of incidence. If P and SV waveforms do not overlap in time at the depth of interest, local velocities of the medium are obtained by separate analysis of these events. The apparent velocity of the event (P or SV) is computed from the moveout across several neighboring depth locations. The angle of incidence of the same event is computed from the particle‐motion hodogram within an appropriately chosen time window. Then, the local medium velocity (P wave or S wave depending on the chosen event) is given by the apparent velocity multiplied by the cosine of the angle of incidence. Layer interfaces with reasonably sharp velocity contrasts are efficient P-wave to SV-wave converters, even at moderate angles of incidence. In offset VSP experiments, converted SV waves are generated with varying strengths at practically all depths. Consequently, the converted SV waveforms partially overlap with the direct P waveforms, making the separate event analysis difficult and inaccurate. These overlapping waveforms can be handled properly by modeling the data in a given time window as a superposition of several events. In particular, the downgoing data at each depth level are modeled as a superposition of a P wave and an SV wave, with local P and S velocities, angles of incidence, and waveforms as model parameters. These parameters are then estimated by minimizing the squared error between the observed data and the model‐generated data. The unknown waveforms are eliminated from the minimization problem, leaving only four nonlinear parameters (velocities and angles) for estimation. Once these four parameters are found, least‐squares estimates of waveforms are obtained by evaluating a simple expression.

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The inverse scattering problem for an acoustic medium is considered within the approximate direct inversion framework. As opposed to iterative methods, the direct inversion approach gives an estimate of the medium velocities by operating on the observed scattered field without repeated solutions of the forward problem. Previous solutions to the multidimensional Born velocity inversion problem require either collocated source and receiver arrays or plane wave sources. In both cases an array of point sources is required to collect the proper data for inversion. In this paper, the solution to the single point source problem is derived. It is shown that by extrapolating and imaging the observed scattered field appropriately, the projections of the velocity function at all angles can be obtained. The velocities are, then, reconstructed by the inverse Radon transform method of tomography.-2- IntroductionThe general inverse scattering problem can be stated as follows. The medium of interest is probed by sources located outside the medium and the scattered field is recorded at various locations. The problem is to obtain the properties of the medium such as the propagation velocity or the material density from the observed data. Similar inverse scattering problems arise in areas such as acoustics, electromagnetics and optics although the corresponding physical phenomenon are somewhat different.In this paper, an acoustic medium probed by sound waves is considered.The medium velocity may vary in all directions, but the density is assumed to be constant. We also assume that the velocity does not depend on frequency; i.e., a non-dispersive medium.The differential equation governing the multidimentional wave propagation in an inhomogeneous medium can be transformed into an integral equaLion known as Lhe Lippiann-Schwinger equalioin from which we obLain the following integral representation of the scattered field, (Taylor [1]), P, (, ) = k2 f dr' 7(1') P(r',c) Go0(r.',w).(1)Here P and PS are the total and scattered fields and Go is the free space Green's function. The velocity function 7 is related to the medium velocities v (r) bywhere 71 (r_) : c ce . where n (.) = (_) and c = ,-is a reference velocity. Since P(r.,c) is also a function of the unknown velocities the scattered field is a nonlinear function of the medium parameters. In the approximate direct inversion approach, the total field in the integrand in equation (1) al.[3] to the case of scattering from a layered medium probed by a point source. Here again the Born inversion can be done with one receiver, and the inversion is obtained via Fourier transforms and a change of variables.Most solutions for the multidimensional case assume .an observation geometry consisting of coincident sources and receivers (zero-offset). One reason for the interest in the zero-offset problem is that for this geometry'the Lippmann-Schwinger integral representation becomes simpler, in that the second and the third terms in the integrand of equation (1) or "fan beam" geometry.As was ment...