The parabolic Radon transform

Seismic Data Analysis

Series	Investigations in Geophysics
Author	Öz Yilmaz
DOI	http://dx.doi.org/10.1190/1.9781560801580
ISBN	ISBN 978-1-56080-094-1
Store	SEG Online Store

An alternative to stretching in the time direction to attain the linear form of the Radon transform (equations 13a, 13b) is given by Hampson . First, the input CMP gather is NMO corrected using the hyperbolic moveout equation

$t_{n}={\sqrt {t^{2}-{\frac {4h^{2}}{v_{n}^{2}}}}},$

(17a)

where t_n is the time after NMO correction, and v_n is the hyperbolic moveout correction velocity function. Resulting moveouts of the events, which were originally hyperbolic, are now approximately parabolic:

$t_{n}=\tau +qh^{2},$

(17b)

where τ is the two-way zero-offset time, and q is the parameter that defines the curvature of the parabola.

In the coordinates of the NMO-corrected gather d(h, t_n), equations (10a, 10b) take the forms

$u(q,\tau )=\sum _{h}d(h,t_{n}=\tau +qh^{2})$

(18a)

and

$d'(h,t_{n})=\sum _{q}u(q,\tau =t_{n}-qh^{2}).$

(18b)

The strategy for computing the hyperbolic Radon transform by performing least-squares minimization for each frequency component of the input data in the stretched coordinates also applies to the moveout-corrected data to compute the parabolic radon transform. Fourier transform the moveout-corrected CMP gather d(h, t_n) in the direction of the moveout correction time variable t_n. Correspondingly, apply Fourier transform to equation (18b) with respect to t_n to obtain

$d'(h,\omega _{n})=\sum _{q}u(q,\omega _{n})\ \exp(-i\omega _{n}qh^{2}),$

(19)

where ω_n is the Fourier dual of t_n.

For each ω_n, define d′ : d′(h, ω_n) and u : u(q, ω_n) as complex vectors in h and q, respectively. With the new definitions of d′ and u, equation (19) can then be written in the matrix form of equation (15), where L now is a complex matrix given by equation (F-29) of Section F.3. The solution that minimizes the least-squares error vector e : e(h, ω_n), now defined in the ω_n domain, accordingly, is given by equation (16).

The complex vectors d′ and u have lengths n_h, the number of offsets, and n_q, the number of constant q values used in the transform defined by equation (19), respectively. The complex matrix L then has dimensions n_h × n_q. For a typical field data set, n_h = 60 and n_q = 60; hence, the complex matrix L may have dimensions of 60 × 60. As for equation (13b), instead of solving one single problem using equation (18b) in the moveout correction time t′ domain that involves a very large matrix, we solve n_ω problems, where n_ω is the number of frequencies ω_n, in the Fourier transform domain using equation (19) involving a small matrix L of equation (F-29).

We now outline the velocity-stack processing based on the discrete parabolic radon transform .

1. Start with a CMP gather, d(h, t) and apply NMO correction, d(h, t_n).
2. Fourier transform in the t_n direction, d(h, ω_n).
3. For each ω_n, set up the L matrix (equation F-29) based on the geometry of the CMP gather and solve for u of equation (16) using the singular-value decomposition (Section F.3).
4. Inverse Fourier transform to get u(q, τ), the parabolic radon transform.
5. Perform a desired operation, such as muting the zone of multiples, in the parabolic radon transform domain.
6. Perform inverse mapping back to the offset domain to get the modeled NMO-corrected CMP gather d′(h, t_n). During this inverse mapping, multiples, primaries, or all of the hyperbolic events can be modeled.
7. Undo the moveout correction to get the modeled CMP gather d′(h, t).

As events on the NMO-corrected CMP gather deviate from the ideal parabolic form, there can be degradation in the ability to map those events into the Radon-transform domain (q, τ) . Moreover, stretch muting that is normally required after NMO correction can remove the far-offset data significantly. The t²-stretching circumvents these shortcomings and replaces the moveout correction of the CMP gather. In practice, both hyperbolic and parabolic schemes in various forms are used to attenuate multiples.

Other equations

$u(v,\tau )=\sum _{h}d(h,t={\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}})$

(10a)

$d'(h,\ t)=\sum _{v}u(v,\tau ={\sqrt {t^{2}-4h^{2}\!/\!v^{2}}})$

(10b)

$u(v,\tau ')=\sum _{h}d(h,t'=\tau '+4h^{2}\!/\!v^{2})$

(13a)

$d'(h,t')=\sum _{v}u(v,\tau '=t'-4h^{2}\!/\!v^{2}).$

(13b)

$\mathbf {d} '=\mathbf {Lu} .$

(15)

$\mathbf {u} =(\mathbf {L^{T\ast }L} )^{-1}\mathbf {L^{T\ast }d} ,$

(16)

See also

• Velocity-stack transformation
• The discrete radon transform
• Practical considerations
• Impulse response of the velocity-stack operator
• Field data examples
• Radon-transform multiple attenuation

External links

find literature about The parabolic Radon transform