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Sjoerd de de Ridder
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Green’s functions and propagator matrices are both solutions of the wave equation, but whereas Green’s functions obey a causality condition in time (G = 0 for t < 0), propagator matrices obey a boundary condition in space. Marchenko-type focusing functions focus a wave field in space at zero time. We discuss the mutual relations between Green’s functions, propagator matrices and focusing functions, avoiding up-down decomposition and accounting for propagating and evanescent waves. We conclude with discussing a Marchenko-type Green’s function representation, which forms a basis for extending the Marchenko method to improve the imaging of steeply dipping flanks and to account for refracted waves.
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Green’s functions and propagator matrices are both solutions of the wave equation, but whereas Green’s functions obey a causality condition in time (G = 0 for t < 0), propagator matrices obey a boundary condition in space. Marchenko-type focusing functions focus a wave field in space at zero time. We discuss the mutual relations between Green’s functions, propagator matrices and focusing functions, avoiding up-down decomposition and accounting for propagating and evanescent waves. We conclude with discussing a Marchenko-type Green’s function representation, which forms a basis for extending the Marchenko method to improve the imaging of steeply dipping flanks and to account for refracted waves.
Standard Marchenko redatuming and imaging schemes neglect evanescent waves and are based on the assumption that decomposition into downgoing and upgoing waves is possible in the subsurface. Recently we have shown that propagator matrices, which circumvent these assumptions, can be expressed in terms of Marchenko focusing functions. In this paper we generalize the relation between the propagator matrix and the Marchenko focusing functions for a 3D inhomogeneous dissipative medium. Moreover, for the same type of medium we discuss a relation between the transfer matrix and the Marchenko focusing functions.
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Standard Marchenko redatuming and imaging schemes neglect evanescent waves and are based on the assumption that decomposition into downgoing and upgoing waves is possible in the subsurface. Recently we have shown that propagator matrices, which circumvent these assumptions, can be expressed in terms of Marchenko focusing functions. In this paper we generalize the relation between the propagator matrix and the Marchenko focusing functions for a 3D inhomogeneous dissipative medium. Moreover, for the same type of medium we discuss a relation between the transfer matrix and the Marchenko focusing functions.
Marchenko redatuming, imaging, monitoring and multiple elimination methods are based on Green’s function representations, with the underlying assumption that the wave field in the subsurface can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. In this paper we show that up/down decomposition in the subsurface is actually not needed for the derivation of these representations. This opens the way for research into new Marchenko methods which are not limited by the assumption that up/down decomposition is possible and which, in principle, can handle evanescent waves.
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Marchenko redatuming, imaging, monitoring and multiple elimination methods are based on Green’s function representations, with the underlying assumption that the wave field in the subsurface can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. In this paper we show that up/down decomposition in the subsurface is actually not needed for the derivation of these representations. This opens the way for research into new Marchenko methods which are not limited by the assumption that up/down decomposition is possible and which, in principle, can handle evanescent waves.
The propagator matrix “propagates” a full wave field from one depth level to another, accounting for all propagation angles and evanescent waves. The Marchenko focusing function forms the nucleus of data-driven Marchenko redatuming and imaging schemes, accounting for internal multiples. These seemingly different concepts appear to be closely related to each other. With this insight, the strong aspects of the propagator matrix (such as the handling of evanescent waves) can be transferred to the focusing function. Vice-versa, the propagator matrix inherits from the focusing function that it can be retrieved from the reflection response, which reduces its sensitivity to the subsurface model.
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The propagator matrix “propagates” a full wave field from one depth level to another, accounting for all propagation angles and evanescent waves. The Marchenko focusing function forms the nucleus of data-driven Marchenko redatuming and imaging schemes, accounting for internal multiples. These seemingly different concepts appear to be closely related to each other. With this insight, the strong aspects of the propagator matrix (such as the handling of evanescent waves) can be transferred to the focusing function. Vice-versa, the propagator matrix inherits from the focusing function that it can be retrieved from the reflection response, which reduces its sensitivity to the subsurface model.
Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.
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Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.
Recently, a novel method to redatum the wavefield in the sub-surface from a reflection response measured at the surface has gained interest for imaging primaries in the presence of strong internal multiples. A prerequisite for the algorithm is an accurate and correct estimate of the direct-wave Green's function. However, usually we use an estimate for the direct-wave Green's function computed in a background velocity medium. Here, we investigate the effect of amplitude and phase errors in that estimate. We formulate two novel imaging conditions based on double-focusing the measured reflection response inside the subsurface. These yield information on the amplitude error in the estimate for the direct-wave Green's function which we can then correct, but the phase error remains elusive.
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Recently, a novel method to redatum the wavefield in the sub-surface from a reflection response measured at the surface has gained interest for imaging primaries in the presence of strong internal multiples. A prerequisite for the algorithm is an accurate and correct estimate of the direct-wave Green's function. However, usually we use an estimate for the direct-wave Green's function computed in a background velocity medium. Here, we investigate the effect of amplitude and phase errors in that estimate. We formulate two novel imaging conditions based on double-focusing the measured reflection response inside the subsurface. These yield information on the amplitude error in the estimate for the direct-wave Green's function which we can then correct, but the phase error remains elusive.