The $\grave{P}\!\acute{P}$ specular reflection coefficient

Now that expressions for the incident and reflected vector wavefields have been obtained in (21) and (24), the least-squares estimation of the $\grave{P}\!\acute{P}$ specular reflection coefficient can be evaluated from (10):

 

$$\grave{P}\!\acute{P}({\bf x};{\bf x}_s) = {\bf {\cal R}}_{11... ...d^2 \left(\Phi^s_1({\bf x},t)\right)^2\,dt + W_m^2({\bf x})}\;.$$ (26)

First, the incident wave scalar can be evaluated given (5) and (21):

 

$$\Phi^s_1({\bf x},t;{\bf x}_s) = {\bf u}^s{\bf \cdot}{\bf e}_1^s = A_o A_s w_1(t-\tau_s) \;.$$ (27)

Next, the reflected wave scalar can be evaluated given (5), (24) and (25):

 

$$\Phi^r_1({\bf x},t;{\bf x}_r) = \u^r{\bf \cdot}{\bf e}_1^r =... ...{\bf \cdot}(\nabla\u){\bf \cdot}{\bf n}'\right]\right\}\,dS'\;.$$ (28)

Note that (28) requires knowledge of $\lambda\nabla{\bf \cdot}\u$, the local normal stress of the displacement field at each receiver, and $\mu\nabla\u$, the local stress tensor at each receiver. For marine seismic data, the pressure $\lambda(\nabla{\bf \cdot}\u)$ is measured directly by hydrophones. However, for both land and marine data acquisition, the local receiver stress tensor is not routinely measured. As suggested by Keho (1986), I use a farfield WKBJ approximation of the displacement vector wavefield gradient and divergence terms such that

 

$$\u({\bf x},t) \sim {\bf U}({\bf x}) e^{iw\tau({\bf x})} \;.$$ (29)

Then  

$$\nabla{\bf \cdot}\u \sim {\bf \dot{\u}}{\bf \cdot}\nabla\tau \;,$$ (30)

and  

$$\nabla\u \sim {\bf \dot{\u}}\nabla\tau \;.$$ (31)

In this case, (28) can be approximated for land acquisition as:

 

$$\Phi^r_1({\bf x},t;{\bf x}_r) = \int_{S'} A_r \left\{ \lam... ...dot{\u}}\nabla\tau_r){\bf \cdot}{\bf n}'\right]\right\}\,dS'\;.$$ (32)

Substituting (27) and (32) into (26) and performing the t integrations, one obtains

 

$${\bf {\cal R}}_{11}({\bf x}) = P_1 / Q_1 \;,$$ (33)

such that

$$\begin{eqnarray} \lefteqn{P_1 =} \nonumber \\ & A_o A_s W_d^2({\bf x},\tau_s) ... ...u}}(t_{sr})\nabla\tau_r){\bf \cdot}{\bf n}'\right]\right\}\,dS'\;,\end{eqnarray}$$ (34)

where ${\bf \dot{\u}}({\bf x},t_{sr})$ is now evaluated at the total ray traveltime $t_{sr}=\tau_s+\tau_r$ from each source at ${\bf x}_s$ to any subsurface point ${\bf x}$ and back up to each receiver position ${\bf x}_r$. The primed coordinates indicate that the polarization vectors should be evaluated at the receiver positions ${\bf x}_r$. The subscripts on $\lambda_r$ and $\mu_r$ indicate that the Lamé parameters should be evaluated at the receiver positions ${\bf x}_r$. Note that the $\ast$ symbol signifies a convolution of the surface data $\u$ with the estimated P-wave source wavelet w₁. The autocorrelation integral is

 

$$Q_1 = A_o^2 A_s^2 \int_{t'}W_d^2({\bf x},t') w_1^2(t')\,dt' + W_m^2({\bf x}) \;,$$ (35)

and is related to the energy of the incident P wavefield. Combining (33)-(35) yields:

$$\begin{eqnarray} \lefteqn{\grave{P}\!\acute{P}({\bf x};{\bf x}_s) = }\nonumber \... ...s \int_{t'}W_d^2({\bf x},t') w_1^2(t')\,dt' + W_m^2({\bf x}) } \;.\end{eqnarray}$$ (36)

Equation (36) gives the least-squares elastic wavefield integral solution for $\grave{P}\!\acute{P}$ specular reflectivity. It can be slightly simplified further by noting that

 

$$\nabla\tau_r = {\bf e}_1^{r'} / \alpha_r \;,$$ (37)

which follows from the P-wave eikonal equation in (15), where $\alpha_r$ is the P-wave velocity evaluated at each receiver location ${\bf x}_r$.