Filippo Broggini

Connection of scattering principles: a visual and mathematical tour

Inverse scattering, Greens function reconstruction, focusing, imaging and the optical theorem are subjects usually studied as separate problems in different research areas. We show a physical connection between the principles because the equations that rule these scattering principles have a similar functional form. We first lead the reader through a visual explanation of the relationship between these principles and then present the mathematics that illustrates the link between the governing equations of these principles. Throughout this work, we describe the importance of the interaction between the causal and anti-causal Greens functions.

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

The methodology of Greens 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 Greens 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 Greens 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 Greens 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.

Data-driven Green's function retrieval and application to imaging with multidimensional deconvolution

An iterative method is presented that allows one to retrieve the Greens function originating from a virtual source located inside a medium using reflection data measured only at the acquisition surface. In addition to the reflection response, an estimate of the travel times corresponding to the direct arrivals is required. However, no detailed information about the heterogeneities in the medium is needed. The iterative scheme generalizes the Marchenko equation for inverse scattering to the seismic reflection problem. To give insight in the mechanism of the iterative method, its steps for a simple layered medium are analyzed using physical arguments based on the stationary phase method. The retrieved Greens wavefield is shown to correctly contain the multiples due to the inhomogeneities present in the medium. Additionally, a variant of the iterative scheme enables decomposition of the retrieved wavefield into its downgoing and upgoing components. These wavefields then enable creation of a ghost-free image of the medium with either cross correlation or multidimensional deconvolution, presenting an advantage over standard prestack migration.

A proposal for model–independent 3D wave field reconstruction from reflection data

With seismic interferometry one can retrieve the response to a virtual source inside an unknown medium, assuming there is a receiver at the position of the virtual source. In a companion paper, Broggini et al. show that for the 1D situation the requirement of having an actual receiver inside the medium can be circumvented. They show that the virtual source response can be obtained from reflection data only. This paper is a first step towards the generalization of their method to the 3D situation. We show how the full response to a virtual source inside the medium can be obtained from the reflection response at the surface. We also indicate how this method can be used to handle internal multiple scattering in migration.

Connection of scattering principles: Focusing the wavefield without source or receiver

Inverse scattering, seismic interferometry, and focusing are subjects usually studied as independent problems in different research areas. We speculate that a physical connection exists between them because the equations that rule these scattering principles have a similar functional form. With a visual explanation of the relationship between these principles, we describe the importance of the interaction between the causal and acausal Green’s functions and provide physical insight to emphasize the connection between these principles. Finally, we show how to reconstruct the Green’s function that radiates from a location where there is no source or receiver, going beyond seismic interferometry.

Source Wavelet Amplitude Spectrum Estimation Using Marchenko Focusing Functions

Summary Marchenko focusing enables the redatuming of seismic surface data to an arbitrary new datum level in the subsurface. One requirement for an accurate redatuming is the correct deconvolution of the source signature from the acquired surface reflection response. Using an erroneous source wavelet for deconvolution leads to inaccurate redatuming and, thus, to artefacts in the redatumed data. However, the true amplitude spectrum of the source wavelet is often unknown. We propose a novel approach to invert for the amplitude spectrum of the source wavelet using focusing functions obtained by the Marchenko redatuming process. Deconvolving the data with an incorrect source wavelet generates undesired coda events in the focusing function. We identify such coda events by the analysis of an ensemble of focusing functions, where differently scaled source wavelets are used for deconvolution prior to redatuming. The amplitude spectrum of the identified coda is strongly related to an error in the source wavelet. Thus, we can iteratively invert for the correct scaling of the source wavelet by analyzing the amplitude spectrum of coda events in the focusing function. This approach yields the correct scaling of the amplitude spectrum of the source wavelet and enables accurate Marchenko redatuming.

Creating Virtual Sources Inside an Unknown Medium from Reflection Data - A New Approach to Internal Multiple Elimination

It has recently been shown that the response to a virtual source in the subsurface can be derived from reflection data at the surface and an estimate of the direct arrivals between the virtual source and the surface. Hence, unlike for seismic interferometry, no receivers are needed inside the medium. This new method recovers the complete wavefield of a virtual source, including all internal multiple scattering. Because no actual receivers are needed in the medium, the virtual source can be placed anywhere in the subsurface. With some additional processing steps (decomposition and multidimensional deconvolution) it is possible to obtain a redatumed reflection response at any depth level in the subsurface, from which all the overburden effects are eliminated. By applying standard migration between these depth levels, a true amplitude image of the subsurface can be obtained, free from ghosts due to internal multiples. The method is non-recursive and therefore does not suffer from error propagation. Moreover, the internal multiples are eliminated by deconvolution, hence no adaptive prediction and subtraction is required.

Data-Driven Green's Function Retrieval from Reflection Data - Theory and Example

Recently we introduced a new approach for retrieving the Greens response to a virtual source in the subsurface from reflection data at the surface. Unlike in seismic interferometry, no receiver is needed at the position of the virtual source. Here we present the theory behind this new method. First we introduce the Greens function G and a so-called fundamental solution F of an inhomogeneous medium. Next we derive a relation between G and F, using reciprocity theorems. This relation is used as the basis for deriving a 3D single-sided Marchenko equation. We show that this equation is solved by a 3D autofocusing scheme and that the Greens function is obtained by combining the focusing wave field and its response in a specific way. We illustrate the method with a numerical example.

Creating the Green's Response to a Virtual Source Inside a Medium Using Reflection Data with Internal Multiples

Seismic interferometry is a technique that allows one to reconstruct the full wavefield originating from a virtual source inside a medium, assuming a receiver is present at the virtual source location. We discuss a method that creates a virtual source inside a medium from reflection data measured at the surface, without needing a receiver inside the medium and, hence, presenting an advantage over seismic interferometry. An estimate of the direct arriving wavefront is required in addition to the reflection data. However, no information about the medium is needed. We illustrate the method with numerical examples in a lossless acoustic medium with laterally-varying velocity and density. We examine the reconstructed wavefield when a macro model is used to estimate the direct arrivals and we take into consideration finite acquisition aperture. Additionally, a variant of the iterative scheme allows us to decompose the reconstructed wave field into downgoing and upgoing fields. These wave fields are then used to create an image of the medium with either crosscorrelation or multidimensional deconvolution.

The contribution of the spatial derivatives to surface-wave interferometry

Summary The effect of near-surface perturbations is still one of the key problems in land seismic surveys. The extraction of direct and scattered ground roll and its removal allowed by interferometry can provide a robust solution for the effects of the shallow geology on surface seismic data. Conventionally, surface waves are estimated implementing a modification of the exact theory of virtual-source interferometry. For example, we usually apply an approximated version of the interferometric equation that does not require spatial derivatives. This introduces errors and spurious events due to the effect of scatterers outside the integration boundary. In this paper we assess whether or not the use of spatial derivatives adds value to the estimates made using virtual-receiver interferometry, using both field experiments and synthetic data.