SUMMARY Seismic interferometry, also known as Green’s function retrieval by crosscorrelation, has a wide range of applications, ranging from surface wave tomography using ambient noise, to creating virtual sources for improved reflection seismology. Despite its successful applications, the crosscorrelation approach also has its limitations. The main underlying assumptions are that the medium is lossless and that the wave field is equipartitioned. These assumptions are in practice often violated: the medium of interest is often illuminated from one side only, the sources may be irregularly distributed, and, particularly for EM applications, losses may be significant. These limitations may partly be overcome by reformulating seismic interferometry as a multidimensional deconvolution (MDD) process. We present a systematic analysis of seismic interferometry by crosscorrelation and by MDD. We show that for the non-ideal situations mentioned above, the correlation function is proportional to a Green’s function with a blurred source. The source blurring is quantified by a so-called point-spread function which, like the correlation function, can be derived from the observed data (i.e., without the need to know the sources and the medium). The source of the Green’s function obtained by the correlation method can be deblurred by deconvolving the correlation function for the point-spread function. This is the essence of seismic interferometry by MDD. We illustrate the crosscorrelation and MDD methods for controlled-source and passive data applications with numerical examples and discuss the advantages and limitations of both methods.

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Seismic interferometry by crosscorrelation and by multidimensional deconvolution: a systematic comparison

Seismic interferometry by crosscorrelation and by multi-dimensional deconvolution: a systematic comparison

Seismic interferometry deals with the generation of new seismic responses by crosscorrelating existing ones. One of the main assumptions underlying most interferometry methods is that the medium is lossless. We develop an ‘interferometry‐by‐deconvolution’ approach which circumvents this assumption. The proposed method applies not only to seismic waves, but to any type of diffusion and/or wave field in a dissipative medium. This opens the way to applying interferometry to controlled‐source electromagnetic (CSEM) data. Interferometry‐by‐deconvolution replaces the overburden by a homogeneous half space, thereby solving the shallow sea problem for CSEM applications. We demonstrate this at the hand of numerically modeled CSEM data.

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Seismic and electromagnetic controlled-source interferometry in dissipative media

An operator formalism is developed to expand the acoustic wave field in a multi‐dimensionally smoothly varying medium, generated by a source localized in space and time, into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times. This expansion is a generalization of the Bremmer coupling series. The condition of smoothness of the medium relates to the width of the signature of the source in the configuration. Both the existence and the convergence (in the weak sense) of the expansion are discussed. The operator calculus involved leads to a natural generalization of the concept of slowness surface to multi‐dimensionally smoothly varying media. The operator associated with the corresponding generalized vertical slowness induces the full one‐way wave operator in the type of media under consideration. In addition, a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents, are derived.

Generalization of the Bremmer coupling series

The methodology of Green’s function retrieval by cross-correlation has led to many interesting applications for passive and controlled-source acoustic measurements. In all applications, a virtual source is created at the position of a receiver. Here a method is discussed for Green’s function retrieval from controlled-source reflection data, which circumvents the requirement of having an actual receiver at the position of the virtual source. The method requires, apart from the reflection data, an estimate of the direct arrival of the Green’s function. A single-sided three-dimensional (3D) Marchenko equation underlies the method. This equation relates the reflection response, measured at one side of the medium, to the scattering coda of a so-called focusing function. By iteratively solving the 3D Marchenko equation, this scattering coda is retrieved from the reflection response. Once the scattering coda has been resolved, the Green’s function (including all multiple scattering) can be constructed from the reflection response and the focusing function. The proposed methodology has interesting applications in acoustic imaging, properly accounting for internal multiple scattering.

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Green's function retrieval from reflection data, in absence of a receiver at the virtual source position

Wave field splitting, invariant imbedding, and phase space methods reformulate the Helmholtz wave propagation problem in terms of an operator scattering matrix characteristic of the modeled environment. The subsequent equations for the reflection and transmission operators are of first-order (one-way) in range, nonlinear (Riccati-like), and, in general, nonlocal. The reflection and transmission operator equations provide the framework for constructing inverse algorithms based on, in principle, exact solution methods.

One-way wave propagation methods in direct and inverse scalar wave propagation modeling

Wave field splitting, invariant imbedding, and phase space methods reformulate the Helmholtz wave propagation problem in terms of an operator scattering matrix characteristic of the modeled environment. The subsequent equations for the reflection and transmission operators are of first-order (one-way) in range, nonlinear (Riccati-like), and, in general, nonlocal. The reflection and transmission operator equations provide the framework for constructing inverse algorithms based on, in principle, exact solution methods

Louis Fishman

Papers

One-way wave propagation methods in direct and inverse scalar wave propagation modeling

One-way wave propagation methods in direct and inverse scalar wave propagation modeling