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Elastic specular reflectivity estimation

Let us outline the mathematical and physical components of the true-amplitude elastic migration reflectivity theory. We refer the reader to the paper of Stolt and Weglein (1985) for a comprehensive overview of seismic migration and inversion techniques.

We begin by considering a small local neighborhood about a single subsurface location ${\bf x_o}$. In general, at ${\bf x_o}$ we can define an incident vector displacement wavefield ${\bf u^{\mbox{\tiny I}}}$ and a scattered vector displacement wavefield ${\bf u^{\mbox{\tiny S}}}$. The wavefield ${\bf u^{\mbox{\tiny I}}}$ is due to the seismic source (dynamite, Vibroseis, airgun array, etc.), and ${\bf u^{\mbox{\tiny S}}}$ contains the scattered reflections and diffractions caused by the source energy bouncing off of rapid changes in subsurface elastic properties, such as reflectors, diffractors and fault zones. The wavefields are vectors since they contain the full three-component directional information of elastic particle displacement, as opposed to directionless measurements of scalar quantities like acoustic pressure. We can define a 3x3 scattering matrix ${\bf R}$ which relates the incident source wavefield to the scattered (reflected or diffracted) wavefield. At reflectors, the elements of the scattering matrix ${\bf R}$ contain the elastic Zoeppritz reflection coefficients $\grave{P}\!\acute{P}$, $\grave{P}\!\acute{S}$, $\grave{S}\!\acute{S}$, etc. Hence, if we can determine any of these coefficients, they will in turn reveal information about the changes in elastic parameters which define them. It is evident that determining these coefficients depends on our ability to reconstruct the source wavefield ${\bf u^{\mbox{\tiny I}}}$ and the scattered wavefield ${\bf u^{\mbox{\tiny S}}}$ locally at each subsurface point ${\bf x_o}$.

To reconstruct the incident and scattered wavefields, we mathematically solve the elastodynamic wave equation which describes the propagation of elastic waves through heterogeneous media (Aki and Richards, 1980). The mathematical technique used is the Kirchhoff (Love) integral solution (Wu, 1989), which gives a way to reconstruct the wavefields in terms of integrals over the seismic source and the recorded seismic trace data at the surface. To evaluate the integrals, we need to calculate the traveltimes and amplitudes of the waves from the source to each subsurface point, $\tau_s$ and As, as well as the traveltimes and amplitudes from each receiver to each subsurface point, $\tau_r$ and Ar (Keho and Beydoun, 1988). We calculate these traveltimes and amplitudes numerically in heterogeneous media by raytracing techniques (Cervený et al., 1977, Beydoun and Keho, 1987).

The final solutions are integrated in a constant offset implementation to preserve the reflection coefficient estimates as a function of fixed specular angles. This is equivalent to presorting the recorded data into constant offset sections, and then migrating each section separately. In practice we do the sorting internal to the migration computer algorithm, so the input data can originate from arbitrary acquisition geometries. Numerically, the integrals become computer summations over all midpoint coordinates ${\bf x_m}$ for a fixed offset ${\bf x_h}$. The midpoint sum for the Rpp coefficient, for a single offset ${\bf x_h}$, is

R_{pp}({\bf x_o};{\bf x_h}) = \sum_{{\bf x_m}} {\bf W}(A_r,A...}\tilde{\u}({\bf x_o},{\bf x_m};{\bf x_h},
 t=\tau_s+\tau_r)\end{displaymath} (1)


R_{pp}({\bf x_o};{\bf x_h}) = \sum_{{\bf x_m}} S_{pp}({\bf x_o},{\bf x_m};{\bf x_h},t=\tau_s+\tau_r)\end{displaymath} (2)

where ${\bf W}$ is a migration weighting function depending on the raytraced source and receiver amplitudes As and Ar, and $\tilde{\u}$ is the recorded trace data which has been deconvolved and filtered. The total factor $S_{pp}= {\bf W}{\bf \cdot}\tilde{\u}$ can be interpreted as a constant offset scattering function, and by summing along the traveltime hyperbolas given by $t=\tau_s+\tau_r$, the summed scatterers make a constant offset reflection.

The specular angles $\theta_{pp}$ can be determined by a first moment summation similar to the Rpp sum,

F(\theta_{pp}({\bf x_o};{\bf x_h})) = 
 \frac{\sum_{{\bf x_m...
 ...\,\, F(\theta_i,\theta_s)} 
 {\sum_{{\bf x_m}} \mid S_{pp}\mid}\end{displaymath} (3)

where F is some known function like $\cos\theta$, and $\theta_i$ and $\theta_s$ are the scattering angles made by the incident ray and scattered ray at each subsurface location. The Rpp and $\theta_{pp}$ results are somewhat similar in spirit to the results of Parsons (1986), which in turn are based on the work of Beylkin (1985) and Bleistein (1987). The main difference is that our method is completely based on the solution to the elastodynamic wave equation in heterogeneous media, as opposed to their acoustic formulations. Several additional theoretical and implementational details further distinguish our work.

So, by implementing the above summations, we obtain estimates of $R_{pp}({\bf x_o},{\bf x_h})$ and $\theta_{pp}({\bf x_o},{\bf x_h})$, which can be combined to obtain $R_{pp}(\theta({\bf x_o}))$; in other words, we can estimate the $\grave{P}\!\acute{P}$ elastic reflection coefficient as a function of specular reflection angle at every point in the subsurface.

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