  Anisotropy of atomic states is characterized by population differences and
coherences between Zeeman sublevels. It can be efficiently created and probed
via resonant interactions with light, the technique which is at the heart of
modern atomic clocks and magnetometers. Recently, nonlinear magneto-optical
techniques have been developed for selective production and detection of higher
polarization moments, hexadecapole and hexacontatetrapole, in the ground states
of the alkali atoms. Extension of these techniques into the range of
geomagnetic fields is important for practical applications. This is because
hexadecapole polarization corresponding to the $\Delta M=4$ Zeeman coherence,
with maximum possible $\Delta M$ for electronic angular momentum $J=1/2$ and
nuclear spin $I=3/2$, is insensitive to the nonlinear Zeeman effect (NLZ). This
is of particular interest because NLZ normally leads to resonance splitting and
systematic errors in atomic magnetometers. However, optical signals due to the
hexadecapole moment decline sharply as a function of magnetic field. We report
a novel method that allows selective creation of a macroscopic long-lived
ground-state hexadecapole polarization. The immunity of the hexadecapole signal
to NLZ is demonstrated with F=2 $^{87}$Rb atoms at Earth's field.
