Quadratic normal moveouts of symmetric reflections in elastic media: A quick tutorial |
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Authors: | M Tygel L T Santos |
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Institution: | (1) Department of Applied Mathematics, IMECC, State University of Campinas (UNICAMP), CP 6065, 13081-970 Campinas (SP), Brazil |
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Abstract: | The design of reflection traveltime approximations for optimal stacking and inversion has always been a subject of much interest
in seismic processing. A most prominent role is played by quadratic normal moveouts, namely reflection traveltimes around
zero-offset computed as second-order Taylor expansions in midpoint and offset coordinates. Quadratic normal moveouts are best
employed to model symmetric reflections, for which the ray code in the downgoing direction coincides with the ray code in
the upgoing direction in reverse order. Besides pure (non-converted) primaries, many multiply reflected and converted waves
give rise to symmetric reflections. We show that the quadratic normal moveout of a symmetric reflection admits a natural decomposition
into a midpoint term and an offset term. These, in turn, can be be formulated as the traveltimes of the one-way normal (N)
and normal-incidence-point (NIP) waves, respectively. With the help of this decomposition, which is valid for propagation
in isotropic and anisotropic elastic media, we are able to derive, in a simple and didactic way, a unified expression for
the quadratic normal moveout of a symmetric reflection in its most general form in 3D. The obtained expression allows for
a direct interpretation of its various terms and fully encompasses the effects of velocity gradients and Earth surface topography. |
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Keywords: | quadratic traveltime NMO symmetric reflections |
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