The parabolic Radon transform
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
where tn is the time after NMO correction, and vn is the hyperbolic moveout correction velocity function. Resulting moveouts of the events, which were originally hyperbolic, are now approximately parabolic:
where τ is the two-way zero-offset time, and q is the parameter that defines the curvature of the parabola.
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, tn) in the direction of the moveout correction time variable tn. Correspondingly, apply Fourier transform to equation (18b) with respect to tn to obtain
where ωn is the Fourier dual of tn.
For each ωn, define d′ : d′(h, ωn) and u : u(q, ωn) 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, ωn), now defined in the ωn domain, accordingly, is given by equation (16).
The complex vectors d′ and u have lengths nh, the number of offsets, and nq, the number of constant q values used in the transform defined by equation (19), respectively. The complex matrix L then has dimensions nh × nq. For a typical field data set, nh = 60 and nq = 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 nω problems, where nω is the number of frequencies ωn, 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 .
- Start with a CMP gather, d(h, t) and apply NMO correction, d(h, tn).
- Fourier transform in the tn direction, d(h, ωn).
- For each ωn, 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).
- Inverse Fourier transform to get u(q, τ), the parabolic radon transform.
- Perform a desired operation, such as muting the zone of multiples, in the parabolic radon transform domain.
- Perform inverse mapping back to the offset domain to get the modeled NMO-corrected CMP gather d′(h, tn). During this inverse mapping, multiples, primaries, or all of the hyperbolic events can be modeled.
- 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.
- Hampson (1986), Hampson, D., 1986, Inverse velocity stacking for multiple elimination: J. Can. Soc. Expl. Geophys., 22, 44–55.
- Velocity-stack transformation
- The discrete radon transform
- Practical considerations
- Impulse response of the velocity-stack operator
- Field data examples
- Radon-transform multiple attenuation