A multi-decadal view of seismic methods for detecting precursors of magma movement and eruption

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.

/pdf/seismic-reflector-imaging-using-internal-multiples-with-1ejidd7v04.pdf

Seismic reflector imaging using internal multiples with Marchenko-type equations

Iterative substitution of the coupled Marchenko equations is a novel methodology to retrieve the Green's functions from a source or receiver array at an acquisition surface to an arbitrary location in an acoustic medium. The methodology requires as input the single-sided reflection response at the acquisition surface and an initial focusing function, being the time-reversed direct wavefield from the acquisition surface to a specified location in the subsurface. We express the iterative scheme that is applied by this methodology explicitly as the successive actions of various linear operators, acting on an initial focusing function. These operators involve multidimensional crosscorrelations with the reflection data and truncations in time. We offer physical interpretations of the multidimensional crosscorrelations by subtracting traveltimes along common ray paths at the stationary points of the underlying integrals. This provides a clear understanding of how individual events are retrieved by the scheme. Our interpretation also exposes some of the scheme's limitations in terms of what can be retrieved in case of a finite recording aperture. Green's function retrieval is only successful if the relevant stationary points are sampled. As a consequence, internal multiples can only be retrieved at a subsurface location with a particular ray parameter if this location is illuminated by the direct wavefield with this specific ray parameter. Several assumptions are required to solve the Marchenko equations. We show that these assumptions are not always satisfied in arbitrary heterogeneous media, which can result in incomplete Green's function retrieval and the emergence of artefacts. Despite these limitations, accurate Green's functions can often be retrieved by the iterative scheme, which is highly relevant for seismic imaging and inversion of internal multiple reflections.

/pdf/on-green-s-function-retrieval-by-iterative-substitution-of-4l9prnl6fi.pdf

On Green’s function retrieval by iterative substitution of the coupled Marchenko equations

The theory of least-squares reverse time migration of multiples (RTMM) is presented. In this method, least squares migration (LSM) is used to image free-surface multiples where the recorded traces are used as the time histories of the virtual sources at the hydrophones and the surface-related multiples are the observed data. For a single source, the entire free-surface becomes an extended virtual source where the downgoing free-surface multiples more fully illuminate the subsurface compared to the primaries. Since each recorded trace is treated as the time history of a virtual source, knowledge of the source wavelet is not required and the ringy time series for each source is automatically deconvolved. If the multiples can be perfectly separated from the primaries, numerical tests on synthetic data for the Sigsbee2B and Marmousi2 models show that least-squares reverse time migration of multiples (LSRTMM) can significantly improve the image quality compared to RTMM or standard reverse time migratio...

Least-squares reverse time migration of multiples

S U M M A R Y Seismic imaging provides much of our information about the Earth’s crustal structure. The principal source of imaging errors derives from simplicistically modeled predictions of the complex, scattered wavefields that interact with each subsurface point to be imaged. A new method of wavefield extrapolation based on inverse scattering theory produces accurate estimates of these subsurface scattered wavefields, while still using relatively little information about the Earth’s properties. We use it for the first time to create real target-oriented seismic images of a North Sea field. We synthesize underside illumination from surface reflection data, and use it to reveal subsurface features that are not present in an image from conventional migration of surface data. To reconstruct underside reflections, we rely on the so-called downgoing focusing function, whose coda consists entirely of transmission-bornmultiple scattering. As such, we provide the first field data example of reconstructing underside reflections with contributions from transmitted multiples, without the need to first locate or image any reflectors in order to reconstruct multiple scattering effects.

Target-oriented Marchenko imaging of a North Sea field

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

Focusing functions are defined as wavefields that focus at a specified location in a heterogeneous subsurface. These functions can be directly related to Green's functions and hence they can be used for seismic imaging of complete wavefields, including not only primary reflections but all orders of internal multiples. Recently, it has been shown that focusing functions can be retrieved from single-sided reflection data and an initial operator (which can be computed in a smooth background velocity model of the subsurface) by iterative substitution of the multidimensional Marchenko equation. In this work, we show that the Marchenko equation can also be inverted directly for the focusing functions. Although this approach is computationally more expensive than iterative substitution, additional constraints can easily be imposed. Such a flexibility might be beneficial in specific cases, for instance when the recorded data are incomplete or when additional measurements (e.g. from downhole receivers) are available.

/pdf/inversion-of-the-multidimensional-marchenko-equation-304oioegk9.pdf

Inversion of the multidimensional marchenko equation

Virtual-source surface wave responses can be retrieved using the crosscorrelation (CC) of wavefields observed at two receivers. Higher mode surface waves cannot be properly retrieved when there is a lack of subsurface sources that excite these wavefields, as is often the case. In this paper, we present a multidimensional-deconvolution (MDD) scheme that is based on an approximate convolution theorem. The scheme introduces an additional processing step in which the CC result is deconvolved by a so-called point-spread tensor. The involved point-spread functions capture the imprint of the lack of subsurface sources and possible anelastic effects, and quantify the associated spatial and temporal smearing of the virtualsource components that leads to the poor surfacewave retrieval. The functions can be calculated from the same wavefields as used in the CC method. For a 2-D example that is representative of the envisaged applications, we show that the deconvolution partially corrects for the smearing. The retrieved virtual-source response only has some amplitude error in the ideal situation of having the depth of the required vertical array equal to the depth penetration of the surface waves. The error is due to ignored cross-mode terms in the approximate convolution theorem. Shorter arrays are also possible. In the limit case of only a single surface receiver, the retrieved virtual-source response is still more accurate than the CC result. The MDD scheme is valid for horizontally layered media that are laterally invariant, and includes exclusively multicomponent point-force responses (rather than their spatial derivatives) and multicomponent observations. The improved retrieval of multimode surface waves can facilitate dispersion analyses in shallow-subsurface inversion problems and monitoring, and surface wave removal algorithms.

/pdf/surface-wave-retrieval-in-layered-media-using-seismic-1gwpsc83xx.pdf

Surface wave retrieval in layered media using seismic interferometry by multidimensional deconvolution

We present a three-dimensional scheme that can be used to compute a vertical radar profile from reflection data measured at the surface. This is done by first constructing a focusing wavefield, which focuses at the chosen location in the subsurface that is then the virtual receiver location for the vertical radar profile Green’s function. Because the upand downgoig parts of the Green’s function are retrieved separately, these are very useful for imaging and inversion. We show with a numerical example that the method works well in a twodimensional configuration.

/pdf/creating-virtual-vertical-radar-profiles-from-surface-366p0yopd0.pdf

C.P.A. Wapenaar

Papers

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

On Green’s function retrieval by iterative substitution of the coupled Marchenko equations

Inversion of the multidimensional marchenko equation

Surface wave retrieval in layered media using seismic interferometry by multidimensional deconvolution

Creating virtual vertical radar profiles from surface reflection ground penetrating radar data