Kirchhoff $\grave{P}\!\acute{P}$ body force equivalent

In this section, we now seek to distinguish the geometric (specular) reflection contributions from the diffraction contributions to the total scattered response derived in (44). Without loss of generality, we define a local geometric/specular/Snell reflection wavefield ${\bf u}^{^{P\!P}}$ such that

 

$${\bf u}^{^{P\!P}}({\bf x};{\bf x}_s) = [{\bf u}^{^{P}}({\bf ... ...;{\bf x}_s)\;\; {\bf \hat{t}}^{^{P\!P}}({\bf x};{\bf x}_s) \;.$$ (45)

The term $\grave{P}\!\acute{P}$ is a geometric reflection coefficient, which may or may not be related to the Zoeppritz plane-wave coefficient, as discussed later. The unit vector ${\bf \hat{t}}^{^{P\!P}}$ is the direction of geometric reflection given by satisfying Snell's Law at the reflecting surface, as sketched in Figure :

 

$${\bf \hat{t}}^{^{P\!P}}= {\bf \hat{t}}^{^{P}}{\bf \cdot}({\bf I}-2{\bf \hat{n}}{\bf \hat{n}}) \;,$$ (46)

where ${\bf \hat{n}}$ is the unit normal to the reflecting surface at ${\bf x}$.Given this definition of ${\bf \hat{t}}^{^{P\!P}}$, the following identity also relates ${\bf \hat{t}}^{^{P}}$ to ${\bf \hat{t}}^{^{P\!P}}$:

 

$${\bf \hat{t}}^{^{P\!P}}{\bf \cdot}{\bf \hat{n}}= - {\bf \hat{t}}^{^{P}}{\bf \cdot}{\bf \hat{n}}\;.$$ (47)

 

raygeom
Figure 1 Scattering geometry.

We derive the following useful properties for ${\bf u}^{^{P\!P}}$:

 

$$\nabla{\bf \cdot}{\bf u}^{^{P\!P}}= \frac{i\omega}{\alpha}{\bf \hat{t}}^{^{P\!P}}{\bf \cdot}{\bf u}^{^{P\!P}}\;,$$ (48)

and

 

$$\nabla{\bf u}^{^{P\!P}}= \frac{i\omega}{\alpha}{\bf \hat{t}}^{^{P\!P}}{\bf u}^{^{P\!P}}\;.$$ (49)

Then, we evaluate the stress field ${\mbox{$\boldmath\sigma$}^{^{P\!P}}}$ associated with this geometric reflectivity as:

 

$${\mbox{$\boldmath\sigma$}^{^{P\!P}}}= \lambda(\nabla{\bf \cd... ...\bf I}+ 2\mu{\bf \hat{t}}^{^{P\!P}}{\bf u}^{^{P\!P}}\right] \;.$$ (50)

Next, we decompose the total stress gradient into background and scattered stress field gradient components, as suggested by the Born analogy and physical interpretation of (36), (37) and (38):

 

$$\nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P}}}= \nabla_o... ...}+ \nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}} \;,$$ (51)

where $\nabla_o$ is the gradient with respect to a smooth (reflectionless) background model, $\nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P}}}$ refers to the direct stress divergence wavefield in the smooth background medium, and $\nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}$ refers to the scattered (reflected) stress divergence wavefield in the smooth background model. There is an implicit and as yet unexplored relationship between the Born model (36) and our Kirchhoff model (51):

 

$$\nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}}^{^{\delta m}} = \nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}\;.$$ (52)

Our Kirchhoff decomposition (51) assumes a linear relationship between the total stress divergence in the true medium versus the direct and scattered stress divergences in the smooth background medium.

We evaluate the scattered stress divergence $\nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}$ as:

 

$$\nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}= \fr... ...ght] + \mbox{O}(\nabla_o\alpha,\nabla_o\lambda,\nabla_o\mu) \;.$$ (53)

The terms of order $(\nabla_o\alpha,\nabla_o\lambda,\nabla_o\mu)$ vanish because the gradient $\nabla_o$ is taken with respect to the smooth reflectionless background medium, and contains no source of scattering. Continuing to evaluate $\nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}$:

$$\begin{eqnarray} \nabla_o{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P\!P}}}& \appro... ... u}^{^{P\!P}}\nonumber \\ & = & -\rho\omega^2{\bf u}^{^{P\!P}}\;,\end{eqnarray}$$ (54)

where we have assumed that

 

$$\nabla_o{\bf u}^{^{P\!P}}\approx \nabla{\bf u}^{^{P\!P}}\;,$$ (55)

which implies that that propagation directions ${\bf \hat{t}}^{^{P\!P}}$, traveltimes $\tau$and spreading amplitudes $A^{^{P\!P}}$ of the scattered wavefield as it propagates from the scattering point to the observation point, are approximately the same in the smooth background as in the true background medium. The divergence of the total stress field in the true medium is then:

 

$$\nabla{\bf \cdot}{\mbox{$\boldmath\sigma$}^{^{P}}}= \nabla_o... ...{^{P\!P}}}= -\rho\omega^2({\bf u}^{^{P}}-{\bf u}^{^{P\!P}}) \;.$$ (56)

The body force equivalent ${\bf f}^{^{P}}$ with respect to ${\bf u}^{^{P\!P}}$ is:

 

$${\bf f}^{^{P}}= -\rho\omega^2{\bf u}^{^{P}}- \nabla{\bf \cdo... ...x{$\boldmath\sigma$}^{^{P}}}= +\rho\omega^2{\bf u}^{^{P\!P}}\;.$$ (57)

Comparing (35) to (57) we see that

 

$${\bf u}^{^{P\!P}}= \frac{\vert{\bf u}^{^{P}}\vert}{i\omega\r... ...)\right] {\bf \cdot}{\bf \hat{t}}^{^{P}}{\bf \hat{t}}^{^{P}}\;,$$ (58)

and since

 

$${\bf \hat{n}}{\bf \cdot}{\bf u}^{^{P\!P}}\equiv {\bf u}^{^{P... ...{P}\!\acute{P}{\bf \hat{t}}^{^{P}}{\bf \cdot}{\bf \hat{n}} \;,$$ (59)

the volumetric body force equivalent geometric reflection coefficient $\grave{P}\!\acute{P}$ can be identified as

 

$$\grave{P}\!\acute{P}= \frac{-1}{i\omega\rho\alpha^2} \left[ ... ...a\mu-\mu\nabla\alpha)\right] {\bf \cdot}{\bf \hat{t}}^{^{P}}\;.$$ (60)

Finally, we calculate the scattered field ${\bf u}^{^{P}}({\bf x}_r)$ in terms of the geometric reflection coefficient $\grave{P}\!\acute{P}$ using (39), (57), (45) and (40):

 

$${\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P}}({\bf x}_r) = \int_{... ..._s{\bf \cdot}{\bf \hat{t}}^{^{P}}_r)\; \,d{\cal V}({\bf x}) \;.$$ (61)

For clarity, we define the angles shown in Figure :

 

$$- {\bf \hat{t}}^{^{P}}({\bf x};{\bf x}_s){\bf \cdot}{\bf \hat{n}}({\bf x}) = \cos\theta_s= \cos\bar{\theta}\;,$$ (62)

 

$${\bf \hat{t}}^{^{P}}({\bf x};{\bf x}_r){\bf \cdot}{\bf \hat{n}}({\bf x}) = \cos\theta_r\;,$$ (63)

 

$${\bf \hat{t}}^{^{P\!P}}({\bf x};{\bf x}_s){\bf \cdot}{\bf \h... ...f x}_r) = \cos\phi_r= \cos(\theta_s+\theta_r-2\bar{\theta}) \;,$$ (64)

and

 

$$- {\bf \hat{t}}^{^{P}}({\bf x};{\bf x}_s){\bf \cdot}{\bf \ha... ... x};{\bf x}_r) = \cos\theta_{sr}= \cos(\theta_s+\theta_r) \;.$$ (65)

 

anglegeom
Figure 2 Generalized reflection ray and angle geometries.

With these definitions, (61) can be written in compact form as:

 

$${\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P}}({\bf x}_r) = \int_{... ..._{sr}} \grave{P}\!\acute{P}\cos\phi_r \,d{\cal V}({\bf x}) \;.$$ (66)

To recap (66), the density at a subsurface point is denoted $\rho({\bf x})$, and the geometric reflection coefficient at that point is $\grave{P}\!\acute{P}({\bf x})$. The amplitude terms A_s and A_r represent the cumulative geometric spreading, transmission loss, Q-attenuation, etc., from the source and receiver to the subsurface point ${\bf x}$ respectively. The factor A_r also includes the vector component projection at the surface location ${\bf x}_r$ as defined by (42) and (43). The term $\tau_{sr}$ is the total traveltime from source at ${\bf x}_s$ to the subsurface point ${\bf x}$ and back up to the receiver at ${\bf x}_r$.Finally, the diffraction weight $\cos\phi_r$ represents the angle between the anticipated geometric specular reflection direction ${\bf \hat{t}}^{^{P\!P}}_s$ and the actual diffraction direction ${\bf \hat{t}}^{^{P}}_r$. In the case of specular reflection when ${\bf \hat{t}}^{^{P\!P}}_s = {\bf \hat{t}}^{^{P}}_r$, $\phi_r=0$ and so $\cos\phi_r=1$.

We now make some general comments about the forward modeling theory given by (66).

• ${\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P}}$ is in fact ${\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P\!P}}$ since we have removed the background field contribution by assuming a smooth background when solving for ${\bf f}^{^{P}}$ in (57).
• The scattered wavefield is linear with respect to $\grave{P}\!\acute{P}$. This follows in that if the local reflector volume is subdivided into multiple smaller volumes, by linear superposition the volume integration result of (66) should not change. If the model is not ray valid for interfaces (i.e., interface topography varies faster than a spatial wavelength of the wavefield), then a linearized version of $\grave{P}\!\acute{P}$ with respect to material property contrasts, such as (60), is required. Otherwise, if interfaces are ray valid, then by extension a $\grave{P}\!\acute{P}$ coefficient which is nonlinearly related to material property contrasts may also be appropriate, such as Zoeppritz.
• $\cos\phi_r$ is symmetric with respect to $\theta_s$ and $\theta_r$ (reciprocity), but not $\grave{P}\!\acute{P}(\theta_s)$.
• It is important to note that the volume integration (66) models a single trace for any arbitrary source and receiver location. There is no ``preferred'' acquisition geometry for this synthesis; it works for shot gathers as well as random trace orientations. In particular, (66) is perfectly adequate for constant offset survey data, which will be the focus of the following inverse theory section.

This completes the forward modeling theory part of our paper that is necessary to proceed onward to the inverse estimation problem.