  We develop methodology for the estimation of the functional mean and the
functional principal components when the functions form a spatial process. The
data consist of curves $X(\mathbf{s}_k;t),t\in[0,T],$ observed at spatial
locations $\mathbf{s}_1,\mathbf{s}_2,...,\mathbf{s}_N$. We propose several
methods, and evaluate them by means of a simulation study. Next, we develop a
significance test for the correlation of two such functional spatial fields.
After validating the finite sample performance of this test by means of a
simulation study, we apply it to determine if there is correlation between
long-term trends in the so-called critical ionospheric frequency and decadal
changes in the direction of the internal magnetic field of the Earth. The test
provides conclusive evidence for correlation, thus solving a long-standing
space physics conjecture. This conclusion is not apparent if the spatial
dependence of the curves is neglected.
