It is well understood that a supercritical superprocess is equal in law to a
discrete Markov branching process whose genealogy is dressed in a Poissonian
way with immigration which initiates subcritial superprocesses. The Markov
branching process corresponds to the genealogical description of prolific
individuals, that is individuals who produce eternal genealogical lines of
decent, and is often referred to as the skeleton or backbone of the original
superprocess. The Poissonian dressing along the skeleton may be considered to
be the remaining non-prolific genealogical mass in the superprocess. Such
skeletal decompositions are equally well understood for continuous-state
branching processes (CSBP). In a previous article, [16], we developed an SDE
approach to study the skeletal representation of CSBPs, which provided a common
framework for the skeletal decompositions of supercritical and (sub)critical
CSBPs. It also helped us to understand how the skeleton thins down onto one
infinite line of descent when conditioning on survival until larger and larger
times, and eventually forever. Here our main motivation is to show the
robustness of the SDE approach by expanding it to the spatial setting of
superprocesses. The current article only considers supercritical
superprocesses, leaving the subcritical case open
