Consider the synthetic CMP gather in Figure 6.4-2c. This gather is a composite of the CMP gather with three primary reflections shown in Figure 6.4-2a and the CMP gather with one primary and its multiples shown in Figure 6.4-2b. Note that the three primaries of the CMP gather in Figure 6.4-2a arrive at the same zero-offset times as the multiples, and the moveout between the primaries and multiples is less than 100 ms at the far offset (2350 m).
Traces in the composite CMP gather (Figure 6.4-2c) are stacked with a range of constant velocities, and the resulting stacked traces are displayed side by side, forming the conventional velocity-stack gather shown in Figure 6.4-2d. Note that the maximum stacked amplitudes correspond to the primary and multiple velocities. The lower-amplitude horizontal streaks in the velocity-stack gather (Figure 6.4-2d) are a result of the contribution of small offsets, while the large-amplitude regions are a result of the contribution of the full range of offsets .
The mapping from the offset domain to the velocity domain is achieved by applying hyperbolic moveout correction and summing over offset given by
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. The input d(h, t) represents the CMP gather, and the output u(v, τ) represents the velocity-stack gather. The inverse mapping from the velocity space back to the offset space is achieved by applying inverse hyperbolic moveout correction and summing over velocity given by
where d′(h, t) represents the modeled CMP gather.
Figure 6.4-2d was obtained by using equation (10a) in which the summation is performed over a finite range of offsets. At first, it appears that by using equation (10b), in which the summation is performed over a range of velocities, the original data d(h, t) in Figure 6.4-2c can be reconstructed from the data in Figure 6.4-2d. The modeled CMP gather d′(h, t) using equation (10b) is shown in Figure 6.4-3b. Observe the relative weakening of amplitudes at far offsets, especially along events with large moveout. Repeated transformations using equations (10a) and (10b) from the velocity domain to the offset domain (Figure 6.4-3) further reduce the amplitudes at far offsets.
Figure 6.4-2 (a) A synthetic CMP gather with three primary reflections; (b) a synthetic CMP gather with one primary reflection (arrival time at 0.2 s at zero-offset time) and its multiples; (c) composite CMP gather containing the primaries and multiples in (a) and (b); (d) the conventional velocity-stack gather derived from the composite CMP gather using equation (10a). Note the amplitude smearing along the velocity axis.
Figure 6.4-3 (a) The same velocity-stack gather as in Figure 6.4-2d; (b) the CMP gather reconstructed from the velocity-stack gather in (a) using equation (10b); (c) velocity-stack gather derived from (b) using equation (10a); (d) CMP gather reconstructed from (c) using equation (10b). Note the degradation of velocity resolution on the velocity-stack gather (c) due to reduction of far-offset amplitudes (b).
Results shown in Figure 6.4-3 demonstrate clearly that the discrete transforms given by equations (10a) and (10b) are not exact inverses of each other. The discrete summation in equation (10a) over a finite range of offsets causes mapping of amplitudes along a hyperbolic event in the offset domain (Figure 6.4-2c) to depart from the ideal point in the velocity domain and results in the smearing of amplitudes along the velocity axis (Figure 6.4-2d). Amplitude smearing means the loss of velocity resolution between two events with little moveout difference.
To understand the inverse problem of restoring the data in the offset domain from the data in the velocity domain, consider the integral forms of equations (10a) and (10b). Reflection times on a CMP gather associated with a horizontally layered earth model can be represented by a Taylor series of the form t = τ + c1h2 + c2h4 + …, where c1, c2, … are scalar coefficients . By including as many terms as desired in the Taylor expansion, the traveltime curve can be expressed by t = τ + ϕ(v, h). This makes the integral form of equation (10a) a special form of the generalized Radon transform  given by
Here, the integration is along curves expressed as linear functions of traveltimes, t and τ. Accordingly, d(h, t) and its Radon transform u(v, τ) are defined as continuous functions in the offset and velocity domains, respectively.
incorporates convolution of u(v, τ) with the rho filter ρ(τ) prior to integration over velocity. In equation (11b), the asterisk denotes convolution. For 2-D data, as for any process that involves summation over a finite spatial aperture (migration principles), the rho filter ρ(τ) has a Fourier transform of the form where ω is the temporal frequency.
- Sherwood and Poe, 1972, Sherwood, J. W. C. and Poe, P. H., 1972, Constant-velocity stack and wavelet processing: Geophysics, 37, 769–787.
- Hubral and Krey, 1980, Hubral, P. and Krey, T., 1980, Interval velocities from seismic reflection time measurements: Soc. Expl. Geophys.
- Beylkin, 1987, Beylkin, G., 1987, Discrete Radon transform: IEEE Trans. on Acoustics, Speech and Signal Processing, AASP-35, 162–172.
- The discrete radon transform
- The parabolic radon transform
- Practical considerations
- Impulse response of the velocity-stack operator
- Field data examples
- Radon-transform multiple attenuation