The least-squares inverse problem is formulated as follows.
Consider our observations consist of a set of constant offset data
.
A constant-offset *l _{2}* misfit energy functional can be defined as

(3) |

Minimizing (3) with respect to and the specular angle leads to two coupled normal equations:

(4) |

where is the spatial wavenumber, and

(5) |

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

(6) |

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

(7) |

In particular,

(8) |

where is the normal to the reflecting surface at the point .The stationary condition (8) is equivalent to:

(9) |

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:

(10) |

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

11/17/1997