The least-squares inverse problem is formulated as follows. Consider our observations consist of a set of constant offset data . A constant-offset l2 misfit energy functional can be defined as
Minimizing (3) with respect to and the specular angle leads to two coupled normal equations:
where is the spatial wavenumber, and
In general, these two coupled equations should be solved simultaneously for and . 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 .In this case, the equation can be solved independently of , and the result can be backsubstituted into the original normal equation for . 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 and . Now I will proceed to solve only the equation under stationary phase, and use the result to solve the equation. Had I started with the assumption of specular reflection (stationary phase), I would have had only one equation for specular , and no equation describing . 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 of the volume integral is
The stationary point of the phase with respect to the integration variable is defined by the equation:
where is the normal to the reflecting surface at the point .The stationary condition (8) is equivalent to:
which in turn is simply stating the law of specular reflection for a 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 . In this case, the misfit error at stationarity reduces to:
Now the error misfit functional (10) is in standard linear form, and can be solved for with a traditional Gauss-Newton gradient optimization method. The solution for should then be backsubstituted into the normal equation, and this decoupling should lead to an l2 solution for , which would not have been otherwise possible had we started directly with the specular form (10).