  We extend the definition of curvature homogeneity of type (1,3) to include
the possibility that there is a homothety between any two points of a manifold
preserving the first r covariant derivatives of the curvature operator
simultaneously; we call this strong curvature homogeneity of type (1,3) up to
order r. We characterize these properties in terms of model spaces. In
addition, we also present two families of three-dimensional Lorentzian metrics
on Euclidean space to exhibit the behavior of this property. The first example
is curvature homogeneous of type (1,3) of all orders, but is not locally
homogeneous. Within this first family, being strongly curvature homogeneous of
type (1,3) up to order 1 implies local homogeneity. The second example is
strongly curvature homogeneous of type (1,3) up to order one, and is not
locally homogeneous, showing that this new definition is not a trivial one.
Within this family, being strongly curvature homogeneous of type (1,3) up to
order 2 implies local homogeneity.
