  It was recently demonstrated that two machine-learning architectures,
reservoir computing and time-delayed feed-forward neural networks, can be
exploited for detecting the Earth's anomaly magnetic field immersed in
overwhelming complex signals for magnetic navigation in a GPS-denied
environment. The accuracy of the detected anomaly field corresponds to a
positioning accuracy in the range of 10 to 40 meters. To increase the accuracy
and reduce the uncertainty of weak signal detection as well as to directly
obtain the position information, we exploit the machine-learning model of
random forests that combines the output of multiple decision trees to give
optimal values of the physical quantities of interest. In particular, from
time-series data gathered from the cockpit of a flying airplane during various
maneuvering stages, where strong background complex signals are caused by other
elements of the Earth's magnetic field and the fields produced by the
electronic systems in the cockpit, we demonstrate that the random-forest
algorithm performs remarkably well in detecting the weak anomaly field and in
filtering the position of the aircraft. With the aid of the conventional
inertial navigation system, the positioning error can be reduced to less than
10 meters. We also find that, contrary to the conventional wisdom, the classic
Tolles-Lawson model for calibrating and removing the magnetic field generated
by the body of the aircraft is not necessary and may even be detrimental for
the success of the random-forest method.
