Reflectivity representation theorems

Consider the elastodynamic wave equation for a displacement vector field $\u_1$ and second order stress tensor field ${\mbox{$\boldmath\sigma$}}_1$ due to a body force vector excitation ${\bf f}_1$:

 

$$-\rho \omega^2 \u_1 = \nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}}_1 + {\bf f}_1 \;,$$ (1)

as per Aki and Richards (1980, p.19, eqn. 2.17). Consider a second displacement and stress field $\u_2$ and ${\mbox{$\boldmath\sigma$}}_2$, with body force ${\bf f}_2$, which also satisfy the elastodynamic wave equation:

 

$$-\rho \omega^2 \u_2 = \nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}}_2 + {\bf f}_2 \;.$$ (2)

The two sets of elastic wavefield can be related by the Divergence Theorem:

 

$$\int_{{\cal V}} \nabla{\bf \cdot}({\mbox{$\boldmath\sigma$}}... ...\u_2 - {\mbox{$\boldmath\sigma$}}_2{\bf \cdot}\u_1)\, d\S \; .$$ (3)

We assume that the scattered (reflected + diffracted) wavefield within the volume ${\cal V}$ is due to a volumetric body force equivalent of reflection surface excitations. This means that, although reflections are created by reflector surface discontinuities in material properties as shown later, we represent those surface excitations as equivalent volumetric body forces, as opposed to conventional surface tractions. In this case, we can write the forward theory explicitly as a mapping from interior points ${\bf x}$ in the volume ${\cal V}$ to an recording surface $\S$. More importantly, the inverse theory is an explicit map from the recorded surface data on $\S$ to the equivalent body force functions at all ${\bf x}$ within ${\cal V}$, rather than to arbitrary reflection surfaces $\cal S'$within ${\cal V}$. This distinction is subtle yet critical for a rigorous solution to the angle-dependent reflectivity inverse problem.

Under the body force equivalence assumption, the surface integral in (3) vanishes (no volume boundary effects):

 

$$\int_{{\cal V}} \nabla{\bf \cdot}({\mbox{$\boldmath\sigma$}}... ...\mbox{$\boldmath\sigma$}}_2{\bf \cdot}\u_1)\,d{\cal V}= 0 \; .$$ (4)

For any vector $\u$ and second order tensor ${\mbox{$\boldmath\sigma$}}$,

 

$$\nabla{\bf \cdot}({\mbox{$\boldmath\sigma$}}{\bf \cdot}\u) =... ...){\bf \cdot}\u + {\mbox{$\boldmath\sigma$}}{\bf :}\nabla\u \;,$$ (5)

where ${\mbox{$\boldmath\sigma$}}{\bf :}\nabla\u$ is a second-order inner contraction of ${\mbox{$\boldmath\sigma$}}$ with $\nabla\u$, and is equivalent to their dot product in this specific case. Using identity (5), (4) can be expanded as:

 

$$\int_{{\cal V}} \left[ (\nabla{\bf \cdot}{\mbox{$\boldmath\s... ...dmath\sigma$}}_2{\bf :}\nabla\u_1 \right] \, d{\cal V} = 0 \;.$$ (6)

We next assume a linear elastic stress-strain relationship such that

 

$${\mbox{$\boldmath\sigma$}}= {\bf C}{\bf :}\nabla\u \;,$$ (7)

where ${\bf C}$ is the fourth-order elastic stiffness tensor C_ijkl. Due to the symmetries inherent in ${\bf C}$ (Aki and Richards, 1980, p.20),

 

$${\mbox{$\boldmath\sigma$}}_1{\bf \cdot}\nabla\u_2 = ({\bf C}... ...nabla\u_1 = {\mbox{$\boldmath\sigma$}}_2{\bf :}\nabla\u_1 \;,$$ (8)

after Frazer and Sen (1985, p.123), for example.

Thus, $({\mbox{$\boldmath\sigma$}}_1{\bf :}\nabla\u_2 - {\mbox{$\boldmath\sigma$}}_2{\bf :}\nabla\u_1)=0$ and so (6) becomes:

 

$$\int_{{\cal V}} \left[ (\nabla{\bf \cdot}{\mbox{$\boldmath\s... ...oldmath\sigma$}}_2){\bf \cdot}\u_1 \right] \, d{\cal V}= 0 \;.$$ (9)

The difference between the two dot-product equations $\u_2{\bf \cdot}\mbox{eqn(\ref{david1/ewe1})}$ and $\u_1{\bf \cdot}\mbox{eqn(\ref{david1/ewe2})}$can be written as

 

$$\u_2{\bf \cdot}(\nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}}_... ..._2) = {\bf f}_2{\bf \cdot}\u_1 - {\bf f}_1{\bf \cdot}\u_2 \;.$$ (10)

Substituting (10) into (9) yields

 

$$\int_{{\cal V}} ({\bf f}_2{\bf \cdot}\u_1 - {\bf f}_1{\bf \cdot}\u_2)\,d{\cal V}= 0\;,$$ (11)

as per Aki and Richards (1980, p.27, eqn. 2.35) with the surface integral terms set to zero. We specify ${\bf f}_2$ such that

 

$${\bf f}_2 = {\bf \hat{a}}_r \delta({\bf x}-{\bf x}_r) \;,$$ (12)

where ${\bf \hat{a}}_r$ selects an arbitrary component of the vector wavefield $\u_2$ at the receiver location ${\bf x}_r$. The delta function form (12) of ${\bf f}_2$ implies that $\u_2$ is by definition a solution to the Green's function problem:

 

$$\rho \omega^2 \u_2 + \nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}}_2 = -{\bf \hat{a}}_r \delta({\bf x}-{\bf x}_r) \;,$$ (13)

such that  

$$\u_2({\bf x};{\bf x}_r) = {\bf \hat{a}}_r{\bf \cdot}{\bf G}({\bf x};{\bf x}_r) \;,$$ (14)

where ${\bf G}$ is the second order Green's tensor. In this case, substituting (12) into (11) and integrating out the delta function reveals:

 

$${\bf \hat{a}}_r{\bf \cdot}\u_1({\bf x}_r) = \int_{{\cal V}} ... ...f}_1({\bf x}){\bf \cdot}\u_2({\bf x};{\bf x}_r)\, d{\cal V}\;.$$ (15)

Equation (15) is our volume integral representation of the reflected wavefield $\u_1$ at ${\bf x}_r$, due to a body force equivalent ${\bf f}_1$ of reflective surface excitation, weighted against the source wavefield $\u_2$ at each subsurface point ${\bf x}$.