The least-squares inverse problem is formulated as follows. Consider our observations consist of a set of constant offset scalar data measured along the receiver component direction. A constant-offset l2 misfit energy functional can be defined as
where . The misfit error energy function (67) attempts to fit the theoretical response to the observed data D, and will give minimum error for the optimal solution of and . Recall that from (66):
Since U is a function of and , which are the quantities we want to estimate from the data D, we minimize (67) with respect to and the geometric specular angle , which leads to two coupled normal equations:
In general, these two coupled equations should be solved simultaneously for and . 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 .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 (66), we have derived two equations, one for each of and . Now we will proceed to solve only the equation under stationary phase, and use the result to solve the equation. Had we started with the assumption of specular reflection (stationary phase), we 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 (67). 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 .Since as in (20), the stationary condition (73) 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 (75) 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 (75).