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 *l _{2}* misfit energy functional can be defined as

(67) |

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):

(68) |

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:

(69) |

and

(70) |

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

(71) |

The stationary point of the phase with respect to the integration variable is defined by the equation:

(72) |

In particular,

(73) |

where is the normal to the reflecting surface at the point .Since as in (20), the stationary condition (73) is equivalent to:

(74) |

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:

(75) |

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 *l _{2}* solution for , which would not have been otherwise possible
had we

11/16/1997