Inverse problem

The least-squares inverse problem is formulated as follows. Consider our observations consist of a set of constant offset data $D({\bf x}_m;{\bf x}_h,w)$. A constant-offset l₂ misfit energy functional can be defined as

 

$$E({\bf x}_h) = \int_w \int_{{\bf x}_m} \left[ D({\bf x}_m;... ...,w) - U({\bf x}_m;{\bf x}_h,w) \right]^2 \,d{\bf x}_m\,dw \;,$$ (3)

Minimizing (3) with respect to $\grave{P}\!\acute{P}$ and the specular angle $\Theta$ leads to two coupled normal equations:

 

$$\frac{\d E}{\d \grave{P}\!\acute{P}} = -2 \int_w \int_{{\bf ... ...s\phi_s \cos\phi_r e^{iw\tau} dV \right] \,d{\bf x}_m\,dw \;,$$ (4)

where $k=w/\alpha$ is the spatial wavenumber, and

 

$$\frac{\d E}{\d \Theta} = -2 \int_w \int_{{\bf x}_m} \left[... ...s\cos\phi_r \right) e^{iw\tau} dV \right] \,d{\bf x}_m\,dw \;.$$ (5)

In general, these two coupled equations should be solved simultaneously for $\grave{P}\!\acute{P}$ and $\Theta$. As that is rather complicated, for now I will present a much simpler approximate approach. The equations can be decoupled by the stationary phase (high-frequency) approximation, in which the major contribution to (2) occurs near the specular point when $\phi_s \approx \phi_r \approx 0$.In this case, the $\grave{P}\!\acute{P}$ equation can be solved independently of $\Theta$, and the result can be backsubstituted into the original normal equation for $\Theta$. It is important to note that by assuming the generalized form of diffraction-reflection in (1), I have derived two equations, one for each of $\grave{P}\!\acute{P}$ and $\Theta$. Now I 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 $\Theta$ equation. Had I started with the assumption of specular reflection (stationary phase), I would have had only one equation for specular $\grave{P}\!\acute{P}$, and no equation describing $\Theta$. 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 (3). The phase component $\gamma$ of the volume integral is

 

$$\gamma({\bf x};{\bf x}_s,{\bf x}_r) = w\tau({\bf x};{\bf x}_... ...u_s({\bf x};{\bf x}_s) + \tau_r({\bf x};{\bf x}_r) \right] \;.$$ (6)

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

 

$$\nabla\gamma = w \nabla\tau = w( \nabla\tau_s + \nabla\tau_r ) = 0 \;.$$ (7)

In particular,

 

$$\nabla\tau_s({\bf x}) {\bf \cdot}{\bf n}({\bf x}) = - \nabla\tau_r({\bf x}) {\bf \cdot}{\bf n}({\bf x}) \;,$$ (8)

where ${\bf n}$ is the normal to the reflecting surface at the point ${\bf x}$.The stationary condition (8) is equivalent to:

 

$$\cos\theta_s({\bf x}) = \cos\theta_r({\bf x}) \;,$$ (9)

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 $\phi_s \approx \phi_r \approx 0$. In this case, the misfit error at stationarity reduces to:

 

$$E({\bf x}_h) = \int_w \int_{{\bf x}_m} \left[ D({\bf x}_m;{... ...ave{P}\!\acute{P}e^{iw\tau} \,dV \right]^2 \,d{\bf x}_m\,dw \;.$$ (10)

Now the error misfit functional (10) 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 $\Theta$ normal equation, and this decoupling should lead to an l₂ solution for $\Theta$, which would not have been otherwise possible had we started directly with the specular form (10).