AMPLITUDE EXTRACTION

Reflection amplitudes at the interface are retrieved by operating in the $\tau-p$ domain. Looking at the recorded time domain CMP records ( Figures and ) we see an amplitude estimation problem near 1.75 sec caused by triplicating branches of reflection hyperbolae. Transforming those records to the $\tau-p$ domain gets rid of triplications and provides a robust estimate of the ``phase domain'' representation of the reflected waves in an anisotropic medium. Thus the traveltime curves in the $\tau-p$ domain are simple curves. From a practical point of view the slant-stack domain offers more advantages such as, dealiasing Nichols (1992), aperture and geometry compensated estimates of the wave field Kostov (1990), and regridding (inteerpolation) capabilities. The events of interest in Figures and (inline) are the uppermost two elliptical curves corresponding to the PP and PS reflection from the shale/chalk interface. The slant-stack difference sections, obtained by subtracting the crossline slant-stack section from the inline counterpart gives Figure and . The differences are obvious at values close to critical horizontal slowness, and at the triplicating event for a large array of horizontal slowness values.

The first step in amplitude extraction is to transform the CMP gathers into the $\tau$-p space using an ``optimum'' least-squares procedure, aiming to give us an unbiased, amplitude preserving estimate of the wave field. The traveltimes in the slant-stack domain are skewed ellipses in the anisotropic case. Since we are interested in PP and PS reflection, it is useful to write the Double Square Root (DSR) equation in the $\tau$-p domain. The DSR equation

$$\tau = {z_0\over{v_{down}}} \sqrt{(1 - p^2 v_{down}^2(p))} + {z_0\over{v_{up}}} \sqrt{(1 - p^2 v_{up}^2(p))}$$ (1)

is a sum of two skewed square roots. Here the velocity depends on the ray parameter, but may not vary laterally. If v_down is not identical to v_up, we are calculating the traveltime for a mode-converted wave. Amplitudes can then be extracted along a prescribed corridor given by the $\tau$-p traveltime curve in conjunction with the dominant wavelet period. From those extracted values the amplitude at the centroid frequency is picked. This gives us a robust reflection energy estimate for a given definite wave type.

 

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Figure 5 Recorded seismograms from a p-source into an x-component.

 

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Figure 6 Recorded seismogram from a p-source into an x-component.

 

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Figure 7 The CMP gather (Fig. ) is transformed using a least-squares slant-stack.

 

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Figure 8 The CMP gather (Fig. ) is transformed using a least-squares slant-stack.

 

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Figure 9 Subtraction of inline and crossline slant-stacks.

 

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Figure 10 Subtraction of inline and crossline slant-stacks.