  We propose nonparametric estimators for the second-order central moments of
possibly anisotropic spherical random fields, within a functional data analysis
context. We consider a measurement framework where each random field among an
identically distributed collection of spherical random fields is sampled at a
few random directions, possibly subject to measurement error. The collection of
random fields could be i.i.d. or serially dependent. Though similar setups have
already been explored for random functions defined on the unit interval, the
nonparametric estimators proposed in the literature often rely on local
polynomials, which do not readily extend to the (product) spherical setting. We
therefore formulate our estimation procedure as a variational problem involving
a generalized Tikhonov regularization term. The latter favours smooth
covariance/autocovariance functions, where the smoothness is specified by means
of suitable Sobolev-like pseudo-differential operators. Using the machinery of
reproducing kernel Hilbert spaces, we establish representer theorems that fully
characterize the form of our estimators. We determine their uniform rates of
convergence as the number of random fields diverges, both for the dense
(increasing number of spatial samples) and sparse (bounded number of spatial
samples) regimes. We moreover demonstrate the computational feasibility and
practical merits of our estimation procedure in a simulation setting, assuming
a fixed number of samples per random field. Our numerical estimation procedure
leverages the sparsity and second-order Kronecker structure of our setup to
reduce the computational and memory requirements by approximately three orders
of magnitude compared to a naive implementation would require.
