The signal covariances *W*_{1}, *W*_{2}, and *W* represent the powers of the
radar signal in orthogonal polarizations and the magnitude and phase of the
cross-covariance or correlated power of the two polarizations. They can be
combined into one entity called the coherency matrix *J*,

The coherency matrix can be decomposed into its polarized and unpolarized components, as

Here,

The covariance measurements determine four quantities: *W*_{1}, *W*_{2}, and
|*W*| and
(or, equivalently,
and ).
The decomposed polarization matrices (20) are described by five
quantities (*A*, *B*, *C*, |*D*|, and ]), whose values can be obtained
from the covariance measurements and from the polarization constraint
*BC*
= |*D*|^{2}. This gives

The decomposed values depend only on the sum and difference of

The above quantities have the interpretation that
(*W*_{1} + *W*_{2}) is the
total signal power, (*B* + *C*) is the total polarized power, and 2*A*is the total unpolarized power. The degree of polarization *p* is defined
as the ratio of the polarized power to the total power, namely

(23) |

From (22), the total polarized power is given by

From this one obtains that

The above formulations apply to any pair of orthogonal polarizations, or polarization basis. Different polarization bases are related by unitary transformations; the determinant and trace of

Expression (25) can be rewritten in the form

where is the geometric mean of the orthogonal powers and is their arithmetic mean. Rearranging terms gives

The above is a fundamental result that is found in general treatments of polarization (e.g., Born and Wolf, 1975; Mott, 1986). It relates the degree of polarization

From the fact that
,
it can be shown that

Expressed in terms of the polarization ratio

(29) |

When the polarization ratio

p |
= | (30) |

For non-unity polarization ratios, the means ratio varies as a function, going to zero when all the power is contained in

In the above expressions,
and the means ratio are functions of the
particular basis in which the polarization measurements are made, but *p* is
independent of basis. In an *H*-*V* basis,
,
and

A similar relation would hold in a circular polarization basis. Equating the two would enable the correlation coefficients in the two bases to be related.