The term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations. The first version of this principle was derived by Claerbout ͑1968͒, who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. For an arbitrary 3D inhomogeneous lossless medium it follows from Rayleigh's reciprocity theorem and the principle of time-reversal invariance that the acoustic Green's function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points. The integral is along sources on an arbitrarily shaped surface enclosing these points. No assumptions are made with respect to the diffusivity of the wavefield. The Rayleigh-Betti reciprocity theorem leads to a similar representation of the elastodynamic Green's function. When a part of the enclosing surface is the earth's free surface, the integral needs only to be evaluated over the remaining part of the closed surface. In practice, not all sources are equally important: The main contributions to the reconstructed Green's function come from sources at stationary points. When the sources emit transient signals, a shaping filter can be applied to correct for the differences in source wavelets. When the sources are uncorrelated noise sources, the representation simplifies to a direct crosscorrelation of wavefield observations at two points, similar as in methods that retrieve Green's functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface.

The split-step Fourier method is developed and applied to migrating stacked seismic data in two and three dimensions. This migration method, which is implemented in both the frequency-wavenumber and frequency-space domains, takes into account laterally varying velocity by defining a reference slowness (reciprocal of velocity) as the mean slowness in the migration interval and a perturbation term that is spatially varying. The mean slowness defines a reference vertical wavenumber which is used in the frequency-wavenumber domain to downward continue the data across a depth interval as in constantvelocity phase-shift migration. The perturbation term is used to define a "source" contribution that is taken into account by the application of a second phase shift in the frequency-space domain. Since the method does not include the effects of second and higher order spatial derivatives of the slowness field, the method theoretically is accurate only when there are no rapid lateral slowness variations combined with steep angles of propagation. However, synthetic and real examples indicate that good results are obtained for realistic geologic structures.

We compare two approaches for deriving the fact that the Green's function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions. One approach is based on physical arguments, exploiting the principle of time-reversal invariance of the acoustic wave equation. The other approach is based on Rayleigh's reciprocity theorem. Using a unified notation, we show that the result of the time-reversal approach can be obtained as an approximation of the result of the reciprocity approach.

[1] In this paper we discuss a new type of regularization technique for the nonlinear inverse scattering problem, namely the multiplicative technique. The main advantage is that we do not have to determine the regularization parameter before the inversion process is started. We consider different norms of the total variation as regularization factor. Specifically, we investigate a weighted L 2 -norm, and by using an appropriate updating scheme we show that this multiplicative regularization factor does not increase the nonlinearity of the inversion problem. Numerical examples using synthetic and experimental data demonstrate the robustness of the presented method. INDEX TERMS:0629 Electromagnetics: Inverse scattering; 0644 Electromagnetics: Numerical methods; 0910 Exploration Geophysics: Data processing; KEYWORDS: Nonlinear inversion, inverse profiling, regularization, total variation Citation: van den Berg, P. M., A. Abubakar, and J. T. Fokkema, Multiplicative regularization for contrast profile inversion,

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