The Radon transform
In the previous section, we discussed slant-stack transformation and its application to multiple attenuation. Slant-stack transformation involves the application of linear moveout correction and summation over the offset axis. As a result of this mapping, the offset axis is replaced with the ray-parameter axis. The relationship between the input coordinates (h, t) and the transform coordinates (p, τ) is given by the linear moveout equation
where p is the ray parameter (the slant-stack transform), t is the two-way traveltime, τ is the two-way intercept time at p = 0, and h is the half-offset.
A companion mapping is called velocity-stack transformation which involves application of hyperbolic moveout correction and summation over the offset axis. As a result of this mapping, the offset axis is replaced with the velocity axis. The relationship between the input coordinates (h, t) and the transform coordinates (v, τ) is given by the hyperbolic moveout equation
where t is the two-way traveltime, τ is the two-way zero-offset time, h is the half-offset and v is the stacking velocity.
Refer to Figure 6.4-1 to review the nature of these two transformations. A linear event in the offset domain, such as a refracted arrival R or linear noise G, ideally maps onto a point in the ray-parameter domain. A hyperbolic event such as a primary (P1, P2) or a multiple (M1, M2, M3) is mapped onto an ellipse in the ray-parameter domain. Since we always have a truncated hyperbola, we inevitably would have a truncated ellipse. A fast-velocity hyperbola maps onto a tighter ellipse than a slow-velocity hyperbola.
Multiples are not periodic in the offset domain, even for a horizontally layered earth model; but they are periodic in the ray-parameter domain. Thus we can use the periodicity for predicting and attenuating multiples in the slant-stack domain as was demonstrated in the slant-stack transform.
Again, we refer to Figure 6.4-1 and now turn our attention to velocity-stack transformation. Since the mapping function is hyperbolic, in this case, a hyperbola in the offset domain, such as a primary or a multiple, ideally maps onto a point in the velocity domain. Hence, we are able to distinguish between multiples and primaries in the velocity domain based on velocity discrimination and use this criterion to attenuate multiples.
The ideal circumstances described by Figure 6.4-1 that a linear event in the offset domain maps onto a point in the ray-parameter domain using equation (9a), and a hyperbolic event in the offset domain maps onto a point in the velocity domain using equation (9b) do not hold in reality. Specifically, a conventional velocity-stack gather consists of constant-velocity CMP-stacked traces. It emphasizes the energy associated with the events that follow hyperbolic traveltime trajectories in the CMP gather. A fundamental problem with velocity-stack transformation is that a CMP gather only includes a cable-length portion of a hyperbolic traveltime trajectory. The finite cable length, discrete sampling along the offset axis and the closeness of hyperbolic summation paths at near offsets cause smearing of the stacked amplitudes along the velocity axis. Unless this smearing is removed, inverse mapping from the velocity domain back to the offset domain does not reproduce the amplitudes in the original CMP gather.
The gather resulting from the inverse mapping can be considered as the modeled CMP gather that contains only the hyperbolic events present in the actual CMP gather. A least-squares minimization of the energy contained in the difference between the actual CMP gather and the modeled CMP gather removes smearing of amplitudes on the velocity-stack gather and increases velocity resolution. A practical application of this procedure is in the separation of multiples from primaries.
- Introduction to noise and multiple attenuation
- Multiple attenuation in the CMP domain
- Frequency-wavenumber filtering
- The slant-stack transform
- Linear uncorrelated noise attenuation
- Multichannel filtering techniques for noise and multiple attenuation