The Kirchhoff elastodynamic integral solution

To evaluate the least-squares reflectivity solution (10), one requires representations for the vector wavefields ${\bf u}^s$ and $\u^r$.

A vector wavefield $\u$ must satisfy the elastodynamic wave equation operator ${\bf {\cal L}}$:

$${\bf {\cal L}}\u = \rho {\bf \ddot{\u}}- \nabla{\bf \cdot}\left( {\bf {\cal C}}{\bf :}\nabla\u \right) = {\bf f}\;,$$

subject to initial and boundary conditions:

 

$$\begin{array} {lcl} \u, \nabla\u & ; & {\bf x}\in S \\ \le... ...\bf \dot{\u}}\right]_{t=t_o} & ; & {\bf x}\in V \end{array} \;,$$ (11)

where $\rho({\bf x})$ is mass density, ${\bf \ddot{\u}}({\bf x},t)$ is the second time derivative of $\u$, ${\bf {\cal C}}({\bf x})$ is the elastic stiffness tensor C_ijkl, and ${\bf f}({\bf x},t)$ is a body force density. The surface S bounds the solution domain volume V, and the (${\bf :}$) symbol is a second order inner contraction. An integral solution can be obtained for $\u$ using Betti's Theorem (the vector equivalent of Green's Theorem for scalars), and by assuming zero initial conditions:

$$\begin{eqnarray} \u({\bf x},t) & = & \int_{t'} \int_{V'} {\bf f}({\bf x}',t')\... ...,t;{\bf x}',t')\right] \; {\bf \cdot}{\bf n}' \right\} \; dS' dt'\end{eqnarray}$$ (12)

as demonstrated in Aki and Richards (1980, p.29). The unit normal to the surface S is denoted as ${\bf n}$. Note that the integral solution (12) is the elastodynamic vector equivalent of the Kirchhoff-Rayleigh-Sommerfeld integral equation for scalar waves.