  This paper addresses the estimation of the second-order structure of a
manifold cross-time random field (RF) displaying spatially varying Long Range
Dependence (LRD), adopting the functional time series framework introduced in
Ruiz-Medina (2022). Conditions for the asymptotic unbiasedness of the
integrated periodogram operator in the Hilbert-Schmidt operator norm are
derived beyond structural assumptions. Weak-consistent estimation of the
long-memory operator is achieved under a semiparametric functional spectral
framework in the Gaussian context. The case where the projected manifold
process can display Short Range Dependence (SRD) and LRD at different manifold
scales is also analyzed. The performance of both estimation procedures is
illustrated in the simulation study, in the context of multifractionally
integrated spherical functional autoregressive-moving average (SPHARMA(p,q))
processes.
