{"id":311,"date":"2012-12-23T07:51:12","date_gmt":"2012-12-23T07:51:12","guid":{"rendered":"http:\/\/ahay.org\/blog\/?p=311"},"modified":"2015-09-02T14:47:45","modified_gmt":"2015-09-02T14:47:45","slug":"program-of-the-month-sfhalfint","status":"publish","type":"post","link":"https:\/\/ahay.org\/blog\/2012\/12\/23\/program-of-the-month-sfhalfint\/","title":{"rendered":"Program of the month: sfhalfint"},"content":{"rendered":"

sfhalfint<\/a> implements half-order integration or differentiation, a filtering operation common in 2-D imaging operators such as as slant stacking or Kirchhoff migration. <\/p>\n

By default, sfhalfint<\/strong> performs half-order integration. To apply half-order differentiation, use inv=y<\/strong>. To apply the adjoint operator, use adj=y<\/strong>. <\/p>\n

Theoretically, half-order integration and differiation correspond to division by $(i\\omega)^{1\/2}$. For stability, $i\\omega$ is replaced in practice by $1-\\rho\\,Z$ when doing differentiation and by $\\frac{1}{2}\\,\\frac{1-\\rho\\,Z}{1+\\rho\\,Z}$ when doing integration. Here $Z=e^{i\\omega\\,\\Delta t}$. As explained by Jon Claerbout<\/a>, this approximation attenuates high frequencies and assures a causal impulse response. The value of the $\\rho$ parameter is controlled by rho=<\/strong>. <\/p>\n

The following plot from bei\/ft1\/hankel<\/a> shows the impulse response of half-order differentiation (also known as the “rho filter”) <\/p>\n

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10 previous programs of the month<\/h3>\n