• journal_title：Geophysics
• Contributor：Jingwei Hu ; Sergey Fomel ; Laurent Demanet ; Lexing Ying
• Publisher：Society of Exploration Geophysicists
• Date：2013-07-01
• Format：text/html
• Language：en
• Identifier：10.1190/geo2012-0240.1
• journal_abbrev：Geophysics
• issn：0016-8033
• volume：78
• issue：4
• firstpage：U41
• section：Seismic Velocity/Statics

Generalized Radon transforms, such as the hyperbolic Radon transform, cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We have devised a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank approximation of the kernel function. The overall structure follows the Fourier integral operator butterfly algorithm. For 2D data, the algorithm runs in complexity <mml:math display="inline"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo> </mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mrow><mml:mo> </mml:mo></mml:mrow><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>O(N2 log N), where <mml:math display="inline"><mml:mi>N</mml:mi></mml:math>N depends on the maximum frequency and offset in the data set and the range of parameters (intercept time and slowness) in the model space. From a series of studies, we found that this algorithm can be significantly more efficient than the conventional time-domain integration.