  The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum
as the matroid analogue of the tree-depth of graphs. The
contraction-deletion-depth, another tree-depth like parameter of matroids, is
the number of recursive steps needed to decompose a matroid by contractions and
deletions to single elements. Any matroid with contraction-deletion-depth at
most d has branch-depth at most d. However, the two notions are not
functionally equivalent as contraction-deletion-depth of matroids with
branch-depth two can be arbitrarily large.
  We show that the two notions are functionally equivalent for representable
matroids when minor closures are considered. Namely, an F-representable matroid
has small branch-depth if and only if it is a minor of an F-representable
matroid with small contraction-deletion-depth. This implies that any class of
F-representable matroids has bounded branch-depth if and only if it is a
subclass of the minor closure of a class of F-representable matroids with
bounded contraction-deletion-depth.
