Abstract
We show that if seismic data <b>d</b> is related to the migration image by <b>m</b><sub><i>mig</i></sub> = <b>L</b><sup>T</sup><b>d</b>. then <b>m</b><sub><i>mig</i></sub> is a blurred version of the actual reflectivity distribution <b>m</b>, i.e., <b>m</b><sub><i>mig</i></sub> = (<b>L</b><sup>T</sup><b>L</b>)<b>m</b>. Here <b>L</b> is the acoustic forward modeling operator under the Born approximation where <b>d</b> = <b>Lm</b>. The blurring operator (<b>L</b><sup>T</sup><b>L</b>), or point spread function, distorts the image because of defects in the seismic lens, i.e., small source-receiver recording aperture and irregular/coarse geophone-source spacing. These distortions can be partly suppressed by applying the deblurring operator (<b>L</b><sup>T</sup><b>L</b>)<sup>-1 </sup>to the migration image to get <b>m</b> = (<b>L</b><sup>T</sup><b>L</b>)<sup>-1</sup><b>m</b><sub><i>mig</i></sub>. This deblurred image is known as a least squares migration (LSM) image if (<b>L</b><sup>T</sup><b>L</b>)<sup>-1</sup><b>L</b><sup>T</sup> is applied to the data <b>d</b> using a conjugate gradient method, and is known as a migration deconvolved (MD) image if (<b>L</b><sup>T</sup><b>L</b>)<sup>-1 </sup>is directly applied to the migration image <b>m</b><sub><i>mig</i></sub> in (<i>k<sub>x</sub>, k<sub>y</sub>, z</i>) space. The MD algorithm is an order-of-magnitude faster than LSM, but it employs more restrictive assumptions. We also show that deblurring can be used to filter out coherent noise in the data such as multiple reflections. The procedure is to, e.g., decompose the forward modeling operator into both primary and multiple reflection operators <b>d</b> = (<b>L</b><sub><i>prim</i></sub> + <b>L</b><sub><i>multi</i></sub>)<b>m</b>, invert for <b>m</b>, and find the primary reflection data by <b>d</b><sub><i>prim</i></sub> = <b>L</b><sub><i>prim</i></sub><b>m</b>. This method is named least squares migration filtering (LSMF). The above three algorithms (LSM, MD and LSMF) might be useful for attacking problems in optical imaging.
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