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The inverse problem

The least-squares inverse problem is formulated as follows. Consider our observations consist of a set of constant offset scalar data $ D({\bf x}_m;{\bf x}_h,\omega)$ measured along the ${\bf \hat{a}}_r$ receiver component direction. A constant-offset l2 misfit energy functional can be defined as

E({\bf x}_h) = \frac{1}{2}
 \int_{\omega} \int_{{\bf x}_m} \...
 ...\bf x}_m;{\bf x}_h,\omega) 
 \right]^2 \,d{\bf x}_m\,d\omega\;,\end{displaymath} (67)

where $U={\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P}}$. The misfit error energy function (67) attempts to fit the theoretical response ${\bf \hat{a}}_r{\bf \cdot}{\bf u}^{^{P}}$ to the observed data D, and will give minimum error for the optimal solution of $\grave{P}\!\acute{P}$ and $\bar{\theta}$. Recall that from (66):

U({\bf x}_m;{\bf x}_h,\omega) = {\bf \hat{a}}_r{\bf \cdot}{\...
 \grave{P}\!\acute{P}\cos\phi_r \,d{\cal V}({\bf x}) \;.\end{displaymath} (68)

Since U is a function of $\grave{P}\!\acute{P}$ and $\bar{\theta}$, which are the quantities we want to estimate from the data D, we minimize (67) with respect to $\grave{P}\!\acute{P}$ and the geometric specular angle $\bar{\theta}$, which leads to two coupled normal equations:

\frac{\d E}{\d \grave{P}\!\acute{P}} = - \int_{\omega} \int_...
 ...phi_r e^{i\omega\tau_{sr}} dV \right] \,d{\bf x}_m\,d\omega\;, \end{displaymath} (69)


\frac{\d E}{\d \bar{\theta}} = 
 -2 \int_{\omega} \int_{{\bf...
 e^{i\omega\tau_{sr}} dV \right] \,d{\bf x}_m\,d\omega\;.\end{displaymath} (70)

In general, these two coupled equations should be solved simultaneously for $\grave{P}\!\acute{P}$ and $\bar{\theta}$. As that is rather complicated, we now present a much simpler approximate approach. The equations can be decoupled by the stationary phase (high-frequency) approximation, in which the major contribution to (68) occurs near the specular point when $\phi_r \approx 0$.In this case, the $\grave{P}\!\acute{P}$ equation can be solved independently of $\bar{\theta}$, and the result can be backsubstituted into the original normal equation for $\bar{\theta}$. It is important to note that by assuming the generalized form of diffraction-reflection in (66), we have derived two equations, one for each of $\grave{P}\!\acute{P}$ and $\bar{\theta}$. Now we will proceed to solve only the $\grave{P}\!\acute{P}$ equation under stationary phase, and use the $\grave{P}\!\acute{P}$ result to solve the $\bar{\theta}$ equation. Had we started with the assumption of specular reflection (stationary phase), we would have had only one equation for specular $\grave{P}\!\acute{P}$, and no equation describing $\bar{\theta}$! That is a very important distinction.

The first step in the decoupling is to apply the method of stationary phase to the volume integral within the misfit error functional E of (67). The phase component $\gamma$ of the volume integral is

\gamma({\bf x};{\bf x}_s,{\bf x}_r) = \omega\tau_{sr}({\bf x...
 ...u_s({\bf x};{\bf x}_s) + \tau_r({\bf x};{\bf x}_r) \right] \;. \end{displaymath} (71)

The stationary point of the phase with respect to the integration variable ${\bf x}$ is defined by the equation:

\nabla\gamma = \omega\nabla\tau_{sr}= \omega( \nabla\tau_s + \nabla\tau_r ) = 0 \;.\end{displaymath} (72)

In particular,

\nabla\tau_s({\bf x}) {\bf \cdot}{\bf \hat{n}}({\bf x}) = - \nabla\tau_r({\bf x}) {\bf \cdot}{\bf \hat{n}}({\bf x}) \;,\end{displaymath} (73)

where ${\bf \hat{n}}$ is the normal to the reflecting surface at the point ${\bf x}$.Since $\nabla\tau={\bf \hat{t}}^{^{P}}/\alpha$ as in (20), the stationary condition (73) is equivalent to:

\cos\theta_s({\bf x}) = \cos\theta_r({\bf x}) \;,\end{displaymath} (74)

which in turn is simply stating the law of specular reflection for a $\grave{P}\!\acute{P}$ wave at a reflecting boundary. In other words, the stationary point, and hence the major contribution to the integral of the misfit error functional, occurs at the condition of specular reflection, when $\phi_r \approx 0$. In this case, the misfit error at stationarity reduces to:

E({\bf x}_h) \approx \frac{1}{2}
 \int_{\omega} \int_{{\bf x...
 ...{P}e^{i\omega\tau_{sr}} \,dV \right]^2 \,d{\bf x}_m\,d\omega\;.\end{displaymath} (75)

Now the error misfit functional (75) is in standard linear form, and can be solved for $\grave{P}\!\acute{P}$ with a traditional Gauss-Newton gradient optimization method. The solution for $\grave{P}\!\acute{P}$ should then be backsubstituted into the $\bar{\theta}$ normal equation, and this decoupling should lead to an l2 solution for $\bar{\theta}$, which would not have been otherwise possible had we started directly with the specular form (75).

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