• journal_title：Geophysics
• Contributor：Yunyue Li ; Yang Zhang ; Jon Claerbout
• Publisher：Society of Exploration Geophysicists
• Date：2012-
• Format：text/html
• Language：en
• Identifier：10.1190/geo2011-0099.1
• journal_abbrev：Geophysics
• issn：0016-8033
• volume：77
• issue：1
• firstpage：V1
• section：Signal Processing

We have developed a hyperbolic penalty function for image estimation. The center of a hyperbola is parabolic like that of an <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>l2 norm fitting. Its asymptotes are similar to <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>l1 norm fitting. A transition threshold must be chosen for regression equations of data fitting and another threshold for model regularization. We combined two methods: Newton’s and a variant of conjugate gradient method to solve this problem in a manner we call the hyperbolic conjugate direction (HYCD) method. We tested examples of (1) velocity transform with strong noise (2) migration of aliased data, and (3) blocky interval velocity estimation. For the linear experiments we performed in this study, nonlinearity is introduced by the hyperbolic objective function, but the convexity of the sum of the hyperbolas assures the convergence of gradient methods. Because of the sufficiently reliable performance obtained on the three mainstream geophysical applications, we expect the HYCD solver method to become our default method.