Once we have obtained the elastic reflectivity, , we can consider how to invert for three elastic parameters at each subsurface location.

We begin by making the linear Bortfeld approximation
(Bortfeld, 1961, p. 489) to the
nonlinear Zoeppritz *R*_{pp} coefficient (Aki and Richards, 1980, p. 153).
This means that we assume that
changes in elastic parameters are *linearly* related to reflectivity.
In practice, this is a good assumption for elastic parameter changes
less than about 20% and specular angles less than about 30
( is a normal incidence reflection). By elastic parameters,
we mean the *change in elastic property divided by the average
elastic property* at any given location. In other words, the parameters
we estimate are *relative changes* and are *not* absolute
magnitudes of properties. Hence we will quote dimensionless values of the
parameters, like +10%, instead of +0.2 g/cc (density) or +300 m/s
(velocity).

The linearized Bortfeld approximation to can be written as

(4) |

where the are known coefficients of specular angle
(mathematically determined by choice of elastic parameterization),
and is the set of three elastic parameters to be found. Generally, *P* is
related to compressional wave properties of the earth, *S* to shear
wave properties, and *D* to density. At a single location we have
*R*_{pp} defined for several values: as many as there
are offsets in the recorded data (recall the constant offset summations).
The Bortfeld approximation above can then be written as a linear
matrix equation

(5) |

where is the matrix of the coefficients ,
is the parameter vector , and
is the corresponding
vector of *R*_{pp} values,

(6) |

If our original data contain 60 offsets (as
in a 120 trace per shot standard marine survey), then we will have 60
values, will be a 60x3 matrix, will be the
3x1 parameter vector, and will be the 60x1 *R*_{pp} (data) vector.
This is clearly an overdetermined problem since we are using 60 data
equations to solve for only 3 parameters. Hence we solve the matrix
solution by least squares with a technique known as Singular Value
Decomposition (SVD),
which decomposes the system into an orthogonal
reference frame and reconstructs an optimal solution by eigenvector
methods (Lines and Treitel, 1984).
We use the spectrum of the 3 eigenvalues to determine the
stability of the solution and to estimate the confidence in the
inversion result, as discussed later.

12/18/1997