Abstract
ABSTRACTA focussing function is a specially constructed field that focusses on to a purely downgoing pulse at a specified subsurface position upon injection into the medium. Such focussing functions are key ingredients in the Marchenko method and in its applications such as retrieving Green's functions, redatuming, imaging with multiples and synthesizing the response of virtual sources/receiver arrays at depth. In this study, we show how the focussing function and its corresponding focussed response at a specified subsurface position are heavily influenced by the aperture of the source/receiver array at the surface. We describe such effects by considering focussing functions in the context of time‐domain imaging, offering explicit connections between time processing and Marchenko focussing. In particular, we show that the focussed response radiates in the direction perpendicular to the line drawn from the centre of the surface data array aperture to the focussed position in the time‐imaging domain, that is, in time‐migration coordinates. The corresponding direction in the Cartesian domain follows from the sum (superposition) of the time‐domain direction and the directional change due to time‐to‐depth conversion. Therefore, the result from this study provides a better understanding of focussing functions and has implications in applications such as the construction of amplitude‐preserving redatuming and imaging, where the directional dependence of the focussed response plays a key role in controlling amplitude distortions.
Highlights
The Marchenko method is a relatively novel technique that relates surface reflection data to Green’s function from any subsurface position based on the concept of focussing function
Rectly modify the input R and f1+,0, it is to be expected that the focussing functions obtained this way for different surface source/receiver array apertures are not the same as those obtained from windowing the focussing functions with a larger source/receiver array aperture as we shall see later in our numerical examples
We obtain mb from mb aperture centre – lateral focal position in time-imaging domain event one-way time × migration velocity and mr directly from the image-ray map. Note that this calculation is done based on Heaviside weighting, where the aperture centre is at the middle of the source/receiver array
Summary
The Marchenko method is a relatively novel technique that relates surface reflection data to Green’s function from any subsurface position based on the concept of focussing function. The total reflection response in this state is R, usually assumed to be represented by the recorded reflection data on the surface Sa, g+ and g− are, respectively, the desired downgoing and upgoing Green’s functions for virtual receivers on S f These can be obtained by solving the time-constrained coupled Marchenko equations based on an initial focussing function from a smooth (migration) velocity model. The corresponding radiation direction in the depth-imaging domain can be obtained after taking into account the additional change in direction from the time-to-depth conversion process This knowledge may lead to key implications in the design of directionally controlled, amplitude-preserving virtual subsurface sources (i.e. redatuming) based on the concept of Marchenko focussing
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