  We examine the radial asymptotic behavior of spherically symmetric
Lemaitre-Tolman-Bondi dust models by looking at their covariant scalars along
radial rays, which are spacelike geodesics parametrized by proper length
$\ell$, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing
quasi-local scalars defined as integral functions along the rays, we obtain a
complete and covariant representation of the models, leading to an initial
value parametrization in which all scalars can be given by scaling laws
depending on two metric scale factors and two basic initial value functions.
Considering regular "open" LTB models whose space slices allow for a diverging
$\ell$, we provide the conditions on the radial coordinate so that its
asymptotic limit corresponds to the limit as $\ell\to\infty$. The "asymptotic
state" is then defined as this limit, together with asymptotic series expansion
around it, evaluated for all metric functions, covariant scalars (local and
quasi-local) and their fluctuations. By looking at different sets of initial
conditions, we examine and classify the asymptotic states of parabolic,
hyperbolic and open elliptic models admitting a symmetry center. We show that
in the radial direction the models can be asymptotic to any one of the
following spacetimes: FLRW dust cosmologies with zero or negative spatial
curvature, sections of Minkowski flat space (including Milne's space), sections
of the Schwarzschild--Kruskal manifold or self--similar dust solutions.
