In order to compare the efficiency of the Snell beam and the transforms for the retrieval of the reflectivity matrix (*R*(*t _{0}*,

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 transform.

The synthetic data, along with its 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 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
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 transform.

*(a)* Snell-beam transform of the CMP gather shown in
Figure **a**, and *(b)* the normal-moveout-corrected
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 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 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**.

1/13/1998