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Impedance estimation

Once I have estimated $\grave{P}\!\acute{P}(\Theta({\bf x}))$, an inverse problem for three isotropic elastic parameters can be posed. Under the assumption that relative contrasts in material properties are small at reflecting boundaries, and the reflection angles are well within the pre-critical region (Aki and Richards, 1980), a linearization of the Zoeppritz plane wave reflection coefficients can be made at every subsurface point ${\bf x}$:

\grave{P}\!\acute{P}(\Theta) \approx 
 I_p c_1(\Theta) + I_s c_2(\Theta) + D c_3(\Theta) \;, \end{displaymath} (15)

where $\{I_p,I_s,D\}$ are the relative contrasts in P impedance, S impedance and density at the reflecting boundary, and $\{c_1, c_2, c_3\}$ are known basis functions which are analytical in $\Theta$. The three basis functions are plotted in Figure [*], with c1 at the top, c2 at the bottom, and c3 near the zero axis in the middle, and are given here analytically as:

c_1 & = & 0.5 \sec^2\Theta \nonumber \\  c_2 & = & -4\gamma^2 \...
 ... \nonumber \\  c_3 & = & 2\gamma^2 \sin^2\Theta - 0.5\tan^2\Theta \end{eqnarray}

where $\gamma$ is the shear to compressional velocity ratio vs/vp. I used a constant value of $\gamma^2 = 0.25$ in this data example, but it could be specified as a function $\gamma({\bf x})$ if the appropriate vp and vs information is available. In fact, the basis functions ci are not too sensitive to reasonable ranges of $\gamma$ values.

In principle, any three elastic parameters $\{P,S,D\}$ can be chosen that span the $\grave{P}\!\acute{P}$ space. I choose the elastic impedance parameterization, because of its robust inversion properties for surface seismic geometries when a narrow (e.g., 5-35) reflection illumination aperture is only available in the data. I explored this issue more completely in Lumley and Beydoun (1991), and it has recently become an active area of discussion in AVO inversion (e.g., De Nicolao et al., 1991).

In particular, I invert (15) at every subsurface location ${\bf x}$by a least-squares method which bootstraps with offset and angle. The logic behind my approach is based on the properties of the basis functions $\{c_1, c_2, c_3\}$, as plotted in Figure [*]. I first find a least-squares estimate for Ip using only $\grave{P}\!\acute{P}$ values for which $\Theta \le 15^{^{\circ}}$. Next, I find a least-squares estimate for Is using the $\grave{P}\!\acute{P}$ data in the range $10^{^{\circ}}\le \Theta \le 40^{^{\circ}}$ and using the estimate of Ip as a constraint on the system. Finally, if there are angles in the data greater than 35, I perform a least-squares estimate for the density parameter using the Ip and Is values as constraints. I have found this method to be a very robust procedure for estimating Ip and Is (e.g., better than damped SVD), and also for demonstrating that little or no independent information on the contribution of density contrasts to a reflection in typical surface seismic geometries is invertible.

Figure 2
Impedance parameterization. Reflectivity unit impulse radiation curves (basis functions) $\{c_1, c_2, c_3\}$ for the impedance parameter set $\{I_p,I_s,D\}$. The uppermost curve is due to a unit perturbation in relative P impedance contrast alone, the lower curve is due to a unit perturbation in relative S impedance contrast alone, and the middle curve is due to a unit perturbation in relative density contrast alone.
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