No abstract

Traditionally, the Marchenko equation forms a basis for 1D inverse scattering problems. A 3D extension of the Marchenko equation enables the retrieval of the Green’s response to a virtual source in the subsurface from reflection measurements at the earth’s surface. This constitutes an important step beyond seismic interferometry. Whereas seismic interferometry requires a receiver at the position of the virtual source, for the Marchenko scheme it suffices to have sources and receivers at the surface only. The underlying assumptions are that the medium is lossless and that an estimate of the direct arrivals of the Green’s function is available. The Green’s function retrieved with the 3D Marchenko scheme contains accurate internal multiples of the inhomogeneous subsurface. Using source-receiver reciprocity, the retrieved Green’s function can be interpreted as the response to sources at the surface, observed by a virtual receiver in the subsurface. By decomposing the 3D Marchenko equation, the response at the virtual receiver can be decomposed into a downgoing field and an upgoing field. By deconvolving the retrieved upgoing field with the downgoing field, a reflection response is obtained, with virtual sources and virtual receivers in the subsurface. This redatumed reflection response is free of spurious events related to internal multiples in the overburden. The redatumed reflection response forms the basis for obtaining an image of a target zone. An important feature is that spurious reflections in the target zone are suppressed, without the need to resolve first the reflection properties of the overburden.

S U M M A R YSeismic 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 wavefield 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 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 interferometric 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.

We present an imaging method that creates a map of reflection coefficients in correct one-way time with no contamination from internal multiples using purely a filtering approach. The filter is computed from the measured reflection response and does not require a background model. We demonstrate that the filter is a focusing wavefield that focuses inside a layered medium and removes all internal multiples between the surface and the focus depth. The reflection response and the focusing wavefield can then be used for retrieving virtual vertical seismic profile data, thereby redatuming the source to the focus depth. Deconvolving the upgoing by the downgoing vertical seismic profile data redatums the receiver to the focus depth and gives the desired image. We then show that, for oblique angles of incidence in horizontally layered media, the image of the same quality as for 1D waves can be constructed. This step can be followed by a linear operation to determine velocity and density as a function of depth. Numerical simulations show the method can handle finite frequency bandwidth data and the effect of tunneling through thin layers.

The one-dimensional Marchenko equation forms the basis for inverse scattering problems in which the scattering object is accessible from one side only. Here we derive a three-dimensional (3D) Marchenko equation which relates the single-sided reflection response of a 3D inhomogeneous medium to a field inside the medium. We show that this equation is solved by a 3D iterative data-driven focusing method, which yields the 3D Green's function with its virtual source inside the medium. The 3D single-sided Marchenko equation and its iterative solution method form the basis for imaging of 3D strongly scattering inhomogeneous media that are accessible from one side only.

A B S T R A C TSeismic 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-bydeconvolution' 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 controlledsource 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.

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.

[1] We analyze the common surface reflection and full-wave inversion methods to retrieve the soil surface dielectric permittivity and correlated water content from airlaunched ground-penetrating radar (GPR) measurements. In the full-wave approach, antenna effects are filtered out from the raw radar data in the frequency domain, and fullwave inversion is performed in the time domain, on a time window focused on the surface reflection. Synthetic experiments are performed to investigate the most critical hypotheses on which both techniques rely, namely, the negligible effects of the soil electric conductivity (s) and layering. In the frequency range 1-2 GHz we show that for s > 0.1 Sm À1 , significant errors are made on the estimated parameters, e.g., an absolute error of 0.10 in water content may be observed for s = 1 Sm À1 . This threshold is more stringent with decreasing frequency. Contrasting surface layering may proportionally lead to significant errors when the thickness of the surface layer is close to one fourth the wavelength in the medium, which corresponds to the depth resolution. Absolute errors may be >0.10 in water content for large contrasts. Yet we show that full-wave inversion presents valuable advantages compared to the common surface reflection method. First, filtering antenna effects may prevent absolute errors >0.04 in water content, depending of the antenna height. Second, the critical reference measurements above a perfect electric conductor (PEC) are not required, and the height of the antenna does not need to be known a priori. This averts absolute errors of 0.02-0.09 in water content when antenna height differences of 1-5 cm occur between the soil and the PEC. A laboratory experiment is finally presented to analyze the stability of the estimates with respect to actual measurement and modeling errors. While the conditions were particularly well suited for applying the common reflection method, better results were obtained using fullwave inversion.