  This paper studies the problem of separating phase-amplitude components in
sample paths of a spherical process (longitudinal data on a unit two-sphere).
Such separation is essential for efficient modeling and statistical analysis of
spherical longitudinal data in a manner that is invariant to any phase
variability. The key idea is to represent each path or trajectory with a pair
of variables, a starting point and a Transported Square-Root Velocity Curve
(TSRVC). A TSRVC is a curve in the tangent (vector) space at the starting point
and has some important invariance properties under the L2 norm. The space of
all such curves forms a vector bundle and the L2 norm, along with the standard
Riemannian metric on S2, provides a natural metric on this vector bundle. This
invariant representation allows for separating phase and amplitude components
in given data, using a template-based idea. Furthermore, the metric property is
useful in deriving computational procedures for clustering, mean computation,
principal component analysis (PCA), and modeling. This comprehensive framework
is demonstrated using two datasets: a set of bird-migration trajectories and a
set of hurricane paths in the Atlantic ocean.
