Fast parabolic transforms

Gilles Darche

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ABSTRACT

I improve here an already existing method of multiples elimination by parabolic transforms. It is based on the approximate parabolic shape of the multiples after NMO correction, and uses a parabolic transform similar in concept to the hyperbolic and slant-stack transforms. This parabolic transform is easy to express in the frequency domain. More important, I will show that its least-squares inverse can also be computed easily, because it requires only the inversion of a Toeplitz matrix. This property makes the transformations especially fast to compute, and is still valid for special cases, like irregular space sampling, or offset-dependent weighting. I will recall the interest of this method for multiple elimination, and extend it to interpolation processes to illustrate the practical advantages of the Toeplitz structure.