The incident wavefield

Given the elastodynamic integral solution (12) and the WKBJ Green's tensor (14), a closed form solution for the incident wavefield ${\bf u}^s({\bf x},t)$ can be derived. I assume the source can be represented as a body point force, and thus evaluated by the volume integral portion only of (12):

 

$${\bf u}^s({\bf x},t) = \int_{t'} \int_{V'} {\bf f}({\bf x}... ...\bf \cdot}\;{\bf {\cal G}}({\bf x},t;{\bf x}',t')\;dV' dt' \;.$$ (17)

A spatially compact impulsive body force may take the general form:

 

$${\bf f}({\bf x}',t') = \left[ A_o w_1(t'){\bf e}_1^{s'} + B_... ...{o3}w_3(t'){\bf e}_3^{s'} \right] \; \d({\bf x}'-{\bf x}_s) \;,$$ (18)

where ${\bf e}_k^{s'}$ are all evaluated at the source location ${\bf x}'={\bf x}_s$.The amplitude $A_o({\bf e}_1^{s'})$ is the scale and radiation pattern of the P source displacement at ${\bf x}_s$, and may vary as a function of the take-off angle which can be obtained from ${\bf e}_1^{s'}$, and in static strength as a function of shot location ${\bf x}_s$.The terms B_o2 and B_o3 are the equivalent S₁ and S₂ factors of A_o. The term $w_1(t';{\bf x}_s)$ is the P waveform, and may vary with source location ${\bf x}_s$. The terms w₂(t') and w₃(t') are the S₁ and S₂ waveforms respectively, and may differ from w₁ in both phase and frequency content as a function of ${\bf x}_s$.

The body force (18) can be substituted into the integral solution (17) for the incident wavefield. The dot product ${\bf f}{\bf \cdot}{\bf {\cal G}}$ is evaluated from (18) and (14), for ${\bf e}_k^n = {\bf e}_k^s$, as

$$\begin{eqnarray} \lefteqn{{\bf f}{\bf \cdot}{\bf {\cal G}}= } \nonumber \\ & & ... ...3}w_3B\d(t-t'-\phi){\bf e}_3^s\right] \d({\bf x}'-{\bf x}_s) \;,\end{eqnarray}$$ (19)

where the remaining polarization vectors ${\bf e}_k^s$ are all evaluated at the observation point ${\bf x}$. Substitution of (19) into (17) and performing the t' integration results in:

 

$${\bf u}^s({\bf x},t) = \int_{V'} \left[ A_o A w_1(t-\tau){\... ...Bw_3(t-\phi){\bf e}_3^s \right] \d({\bf x}'-{\bf x}_s)\,dV' \;.$$ (20)

The final volume integration over V' yields a compact form for the incident wavefield solution:

 

$${\bf u}^s({\bf x},t) = A_o A_s w_1(t-\tau_s){\bf e}_1^s + B... ...(t-\phi_s){\bf e}_2^s + B_{o3}B_s w_3(t-\phi_s){\bf e}_3^s \;,$$ (21)

where the notation A_s means $A({\bf x};{\bf x}_s)$ and $\tau_s$ means $\tau({\bf x};{\bf x}_s)$,i.e., the value at ${\bf x}$ due to a source at ${\bf x}_s$. I remind you again that the polarization vectors ${\bf e}_k^s$ in (21) are to be explicitly evaluated at each subsurface location ${\bf x}$.To evaluate (21), the WKBJ amplitudes and traveltimes $\{A_s,B_s,\tau_s,\phi_s\}$, and the polarization vectors ${\bf e}_k^s({\bf x},t)$, need to be raytraced from each source position ${\bf x}_s$ to each subsurface position ${\bf x}$, by numerically solving systems (15)-(16) with a rayracing or finite difference algorithm.