Consider a point in the subsurface. I define the reflectivity matrix at the point by the following relation between the incident displacement vector field due to a source at , and the reflected displacement vector field due to a receiver at the position :

(1) |

where all quantities are evaluated locally at the point .The reflectivity matrix is of order (3x3) and, in general, is a scattering matrix which relates to and contains refractions, diffractions and reflections. However, by anticipating that will be simulated as a direct primary incident field, and will be reconstructed from refraction-muted surface observations as a single-conversion primary reflected field, the interpretation of as a generalized reflectivity matrix is justified.

Now consider the local plane-wave unit
polarization vectors , where is the local
*P*-wave propagation direction, the *S _{1}* direction,
and the

where *u _{1}* is the

(2) |

The generalized non-specular reflection coefficients are related to the plane-wave specular Zoeppritz coefficients as discussed by Frazer and Sen (1985), and will be examined in more detail later in this paper. It is apparent that the generalized reflection coefficients within can be ``isolated'' by the appropriate vector dot products. For example, the coefficient can be isolated as

(3) |

where is the local *P*-wave polarization vector of the incident
displacement vector , evaluated at the subsurface point ,
and is the
local *P*-wave polarization vector of the reflected displacement vector
, evaluated at the subsurface point .
Please refer to Figure
to see the appropriate polarization vector geometry.
Similar dot product relations can be stated for the other
reflection coefficients
of using the appropriate vector directions .For example, the coefficient can be isolated as

(4) |

I will define the reflected scalar function on the left of (4) as and the incident scalar function on the right of (4) as :

and

(5) |

Figure 1

If I can forward model numerous source wavefields and I have numerous
reconstructions of the reflected wavefield from multiple
surface shot gather observations, then (5)
(and all of its nine converted-wave counterparts)
can be rearranged into an optimization problem to estimate
the isolated reflection coefficient, hereafter called .
The objective function *J* for this problem has the general *l*_{p} form

(6) |

where *W*_{d} and *W*_{m} are arbitrary weighting functions in data and model
space respectively. The weighting functions in this form are nonconvolutional,
and therefore correspond to diagonal covariance functions in the
usual least-squares theory sense.
Note that adding the second term to (6)
will add stability to the final least-squares solution, and will ensure
that will be a minimum energy model in the *l*_{p} sense, which is
appropriate for a desired spiky reflectivity solution.
Choosing the *l _{2}* norm and integrating over the
implicit functional dependencies, one obtains:

(7) |

To extremize *J* and thus find an ``optimal'' solution for , I perform
the following stationary point analysis.
Perturb *J* by an arbitrary small amount :

(8) |

A stationary point of *J* with respect to occurs when

Neglecting terms of order ,

(9) |

Since is *arbitrary*, then the integrand kernel
*must* vanish at the stationary point. This results in a least-squares
reflectivity solution:

(10) |

Equation (10) is a weighted zero-lag correlation of the incident and reflected wavefield scalar products, normalized by the weighted minimum-threshold energy of the incident wavefield scalar product. Equation (10) is valid for all of the generalized elastic reflection coefficients of (2) by simply making the appropriate redefinitions of and as in (5).

11/17/1997