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The property of the angle-dependent
reflectivity matrix *R*(*z*_{n}) in equation ()
becomes clearer if we look at it in the Fourier domain
as de Bruin et al. (1990) does.
In order to obtain an expression for ,we start with the well-known angle-dependent reflection
coefficient for two acoustic half-spaces separated
by an interface at *z*_{n}:

| |
(44) |

where *c*_{1} and *c*_{2} are the velocities,
and are the mass densities
of the upper and lower half-space, respectively,
and is the angle of incidence.
By substituting , , and
into equation () we obtain

| |
(45) |

As an example, Figure shows the angle-dependent
reflection coefficients in the and
domains when *c*_{1}=1500 *m*/*s*, *c*_{2}=3000 *m*/*s*,
, and .Since the reflection is a convolution of the downgoing wave field
with the reflection coefficient,
the reflectivity matrix in equation ()
can be visualized by taking the reflection coefficient for a given
frequency in Figure and making a matrix
whose columns are down-shifted reflection coefficients of
each other.
Figure shows this reflectivity matrix
when .

**rc-wk-wx
**

Figure 1 Angle-dependent reflection coefficients in (left) and (right) domains when *c*_{1}=1500 *m*/*s*, *c*_{2}=3000 *m*/*s*,, and .

**rc-mat
**
Figure 2 Reflectivity matrix *R* for . |
| |

** Next:** Wave field extrapolation into
** Up:** The forward model
** Previous:** Propagation operators
Stanford Exploration Project

2/5/2001