  The bivariate $P$- and $Q$-polynomial structures of association schemes based
on attenuated spaces are examined using recurrence and difference relations of
the bivariate polynomials which form the eigenvalues of the scheme. These
bispectral properties are obtained from contiguity relations of univariate dual
$q$-Hahn and affine $q$-Krawtchouk polynomials. The bispectral algebra
associated to the bivariate polynomials is investigated, as well as the
subconstituent algebra of the schemes. The properties of the schemes are
compared to those of the non-binary Johnson schemes through a limit.
