  Originally complied for 1868-1967 and subsequently continued so that it now
covers 150 years, the $aa$ index has become a vital resource for studying space
climate change. However, there have been debates about the inter-calibration of
data from the different stations. In addition, the effects of secular change in
the geomagnetic field have not previously been allowed for. As a result, the
components of the 'classical' $aa$ index for the southern and northern
hemispheres ($aa_S$ and $aa_N$) have drifted apart. We here separately correct
both $aa_S$ and $aa_N$ for both these effects using the same method as used to
generate the classic $aa$ values but allowing ${\delta}$, the minimum angular
separation of each station from a nominal auroral oval, to vary as calculated
using the IGRF-12 and gufm1 models of the intrinsic geomagnetic field. Our
approach is to correct the quantized aK-values for each station, originally
scaled on the assumption that ${\delta}$ values are constant, with
time-dependent scale factors that allow for the drift in ${\delta}$. This
requires revisiting the intercalibration of successive stations used in making
the $aa_S$ and $aa_N$ composites. These intercalibrations are defined using
independent data and daily averages from 11 years before and after each station
change and it is shown that they depend on the time of year. This procedure
produces new homogenized hemispheric aa indices, $aa_{HS}$ and $aa_{HN}$, which
show centennial-scale changes that are in very close agreement. Calibration
problems with the classic $aa$ index are shown to have arisen from drifts in
${\delta}$ combined with simpler corrections which gave an incorrect temporal
variation and underestimate the rise in $aa$ during the 20th century by about
15%.
