  Constraints on the interior of the Moon have been derived from its inductive
response, principally as measured by the magnetic transfer function (TF)
between the distantly orbiting Explorer 35 satellite and the Apollo 12 surface
station. The most successful prior studies used a dataset spanning 0.01-1 mHz,
so the lunar response could be modeled as a simple dipole. However, earlier
efforts also produced transfer functions up to 40 mHz. The smaller
electromagnetic skin depth at higher frequency would better resolve the
uppermost mantle-where key information about primitive lunar evolution may
still be preserved-but requires a multipole treatment. I compute new profiles
of electrical conductivity vs depth using both low- and high-frequency ranges
of published Apollo-Explorer TFs. Using the low-frequency data, I derive
temperature profiles at depths >400 km (<1 mHz) consistent with conductive heat
loss and expectations of the iron (and possibly water) content of the mantle.
The near-constant iron fraction (Mg# 81 +/- 10) could imply efficient mixing
due to now-defunct convection. Alternatively, incomplete overturn of
gravitationally unstable magma-ocean cumulates could have left a heterogeneous
distribution of minerals at hundred-km scales that are not resolved by
electromagnetic sounding. A third explanation is that the electromagnetically
probed region may be the initial equilibrium crystallization in a mantle that
did not buoyantly overturn. In contrast, the high-frequency data produced
higher conductivities than expected, requiring unrealistically low Mg# or high
water content. Either the published transfer functions >> 1 mHz are incorrect,
or the TF multipole method at the Moon is unreliable. Future electromagnetic
sounding using the magnetotelluric method can operate up to 100s Hz and is
largely insensitive to multipole effects, resolving structure to 100 km or
less.
