Focusing aspects of the hyperbolic Radon transform

Autor: Samuel H. Bickel
Rok vydání: 2000
Předmět:
Zdroj: GEOPHYSICS. 65:652-655
ISSN: 1942-2156
0016-8033
DOI: 10.1190/1.1444762
Popis: The parabolic approximation does not accurately model residual moveout for long-offset marine data. Consequently the focusing power of the parabolic Radon transform is degraded. Maeland (1998) analyzes this problem by deriving the envelope of hyperbolic events in the (τ, q ) domain. This note extends Maeland's analysis to the hyperbolic Radon transform (τ, p ) domain. Following the notation of Maeland (1998) for the parabolic transform, I modify his equation (5) to model the fixed-focus hyperbolic Radon transform domain as defined by Foster and Mosher (1992): ![Formula][1] (1) where t ( x ) is the traveltime curve for the event, p is the slowness parameter that scales the stacking hyperbola, z is the focusing depth of the hyperbola, and x is offset. The focusing depth, z , governs the shape of the stacking curve and hence the nature of the transform. For example, if z = 0, then the hyperbolic term in equation (1) reduces to px , and the transform becomes the linear radon transform. On the other hand, if z → ∞, the hyperbolic term reduces to qx 2, where ![Formula][2] (2) which is the parabolic Radon transform analyzed by Maeland. As described by Maeland, the envelope can be found by solving the simultaneous equations F = ∂ F /∂ x = 0. These conditions lead to the following pair of equations for the hyperbolic Radon transform: ![Formula][3] (3) and ![Formula][4] (4) A hyperbolic event in the ( t , x ) domain can be written as ![Formula][5] (5) where T is the traveltime at x = 0 and ν is the moveout velocity. By substituting the hyperbolic event [equation (5)] into equation (4), I solve for x 2, which when substituted into the equation for τ [equation (3)] yields the following (τ, p ) curve: ![Formula][6] (6) If z = ν T , then the … [1]: /embed/graphic-1.gif [2]: /embed/graphic-2.gif [3]: /embed/graphic-3.gif [4]: /embed/graphic-4.gif [5]: /embed/graphic-5.gif [6]: /embed/graphic-6.gif
Databáze: OpenAIRE
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