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In order to compare the efficiency of the Snell beam and the $\tau-p$ transforms for the retrieval of the reflectivity matrix (R(t0,p)), I generated a synthetic CMP gather, using an elastic modeling program. This program assumes a 1D geology, and is based on Haskel-Tompson propagation matrices. The layers (specified in Table 1) were chosen so that five different reflectivity functions could be obtained, with a constant density and only two different values of P-wave velocities and Poisson's ratio.

Layer P velocity (m/s) Poisson's Ratio Density (g/cc)
  1500 0.5 1.0
1 2300 0.1 2.0
2 3000 0.3 2.0
3 2300 0.1 2.0
4 2300 0.3 2.0
5 3000 0.1 2.0
6 2300 0.3 2.0

Table 467

Elastic constants of the synthetic model

(a) Synthetic CMP gather, with the five reflections of interest labeled (with numbers), and (b) its $\tau-p$ transform.

The synthetic data, along with its $\tau-p$ transform, is shown in Figure [*]. The strong reflection at 0.4 seconds corresponds to the water-bottom interface (with a large impedance contrast). To avoid the presence of multiples corresponding to the air-water boundary, the air-filled upper space was replaced by water in the model. The five primary P reflections of interest are labeled (with numbers) in the figure: reflection 1 corresponds to the interface between layers 1 and 2 on Table 1, and so on. These labels will be used later to refer to each of these reflectors.

The Snell-beam transform of the synthetic data and its normal-moveout-corrected $\tau-p$ transform are shown on Figures [*] a and [*] b. The first evident difference between the two transforms is the absence of information for small values of p, in the Snell-beam transform. Since there are some missing traces close to the source, we should expect the presence of the gap (that decreases with time). However, the $\tau-p$ transform shows a smearing effect, generating information on values of slowness not covered by the experiment. Another visible difference is the relatively greater strength of the ocean bottom multiples on the $\tau-p$ transform.

(a) Snell-beam transform of the CMP gather shown in Figure [*] a, and (b) the normal-moveout-corrected $\tau-p$ transform.

The general amplitude behavior of the five labeled reflections are very similar on both transforms. I chose reflection 5 to compare how closely the two transforms approximate the theoretical plane-wave response. Figures [*] a and b shows a small window of eighteen samples around reflection 5, for both transforms (negative amplitudes are plotted in the up direction). Although more visible on the Snell-beam transform, the phase inversion (zero crossing) is also clear on the $\tau-p$transform.

(a) and (b) Small windows around reflection 5 of Figures [*] a and b respectively, and (c) and (d) the two central samples of these windows (continuous lines) as well as the theoretical curves for the reflection coefficient (dotted line), and the reflection response with the transmission losses considered (dashed line).

The two central time samples of Figures [*] a and b are plotted in Figures [*] c and d (continuous lines), as well as the theoretical curves for the reflection coefficient (dotted lines), and the reflection response with all the transmission losses (including mode conversions) taken into account (dashed lines). Although both transforms show a good fit with the theoretical curve for values of p between the minimum recorded value and the zero-crossing, the $\tau-p$ transform seems to be more susceptible to multiples contamination. For this reason the zero crossing value of p, is better defined in the Snell-driven transform. However, for values of p beyond the zero crossing, the amplitudes of the transformed data are lower than those of the theoretical curve for both transforms.

The same kind of curves, corresponding to reflections 1-4 of the Snell-beam transform, are presented in Figure [*]. Continuous lines correspond to the central time sample of the Snell-beam transform. Dotted lines are the theoretical curves for the reflection coefficient, and dashed lines are the reflection responses with the transmission losses taken into account. In a and d (where the P wave velocity increases) the reflection coefficient becomes imaginary after the critical p, and the imaginary part of the two theoretical curves are also present (dot-dash and small dash).

As in Figure [*]c and d (a), (b), (c) and (d) correspond to reflections 1, 2, 3, and 4 respectively. See further explanation in the text.

In all cases there is a good fit for small values of p (on the range of available information), but for large values of p the transformed amplitudes are always smaller than the theoretical values. The problem arises from the modeling scheme's failure to maintain the same frequency content for the entire range of p. A comparison of the spectra of a near-, intermediate- and far-offset traces for the synthetic data and for real data are shown in Figure [*]. While for real data the amplitude loss is about the same for all offsets, the synthetic data has a considerable decrease of the high frequency content.

(a) Spectra of three traces at near-, intermediate-, and far-offset for a real data, and (b) the same for the synthetic data of Figure [*] a.

Figure [*] shows a shot gather of a marine dataset from Brazil, and its Snell-beam transform. The data was recorded with a long cable (150 receiver-groups, maximum offset of 4.3 Km), and the absence of structural features is in good agreement with the assumption of a plane-layered earth. We can notice some strong events appearing at large values of p as we should expect from reflections near the critical angle. Also, a careful examination of the sharp reflection at 1.25 seconds, on the transformed data, suggests an increase in the reflection coefficient with p. For such large values of p, the hyperbolic approximation for the reflection moveout is not appropriate any more. I used the stacking velocities (found with the use of only pre-critical offsets) and constructed a model for raytracing.

(a) A long cable marine data from Brazil, and (b) its Snell-beam transform. Notice the strong events associated with reflections near the critical angle.

The two central time slices of this reflection are shown in Figure [*] and, as we can notice, the general shape of the curve is very similar to the curve of reflection 1 (Figure [*] a) of the synthetic data. We can infer, even without submitting the data to an inversion algorithm, that the reflection corresponds to an increase in P velocity [*] and an increase in the Poisson's ratio.

Relative amplitudes of the two central time slices of the reflection at 1.25 seconds of Figure [*] b.

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