  We extend formulae which measure discrepancies for regularized traces on
classical pseudodifferential operators to regularized trace cochains,
regularized traces corresponding to 0-regularized trace cochains. This
extension from 0-cochains to $n$-cochains is appropriate to handle
simultaneously algebraic and geometric discrepancies/anomalies. Algebraic
anomalies are Hochschild coboundaries of regularized trace cochains on a fixed
algebra of pseudodifferential operators weighted by a fixed classical
pseudodifferential operator with positive order and positive scalar leading
symbol. In contrast, geometric anomalies arise when considering families of
pseudodifferential operators associated with a smooth fibration of manifolds.
They correspond to covariant derivatives (and possibly their curvature) of
smooth families of regularized trace cochains, the weight being here an
elliptic operator valued form on the base manifold. Both types of discrepancies
can be expressed as finite linear combinations of Wodzicki residues.We apply
the formulae obtained in the family setting to build Chern-Weil type weighted
trace cochains on one hand, and on the other hand, to show that choosing the
curvature of a Bismut-Quillen type super connection as a weight, provides
covariantly closed weighted trace cochains in which case the geometric
discrepancies vanish.
