  We investigate the problem of probing the local spatial structure of the
magnetic field of the interstellar medium using multi-frequency polarized maps
of the synchrotron emission at radio wavelengths. We focus in this paper on the
three-dimensional reconstruction of the largest scales of the magnetic field,
relying on the internal depolarization (due to differential Faraday rotation)
of the emitting medium as a function of electromagnetic frequency. We argue
that multi-band spectroscopy in the radio wavelengths, developed in the context
of high-redshift extragalactic HI lines, can be a very useful probe of the 3D
magnetic field structure of our Galaxy when combined with a Maximum A
Posteriori reconstruction technique. When starting from a fair approximation of
the magnetic field, we are able to recover the true one by using a linearized
version of the corresponding inverse problem. The spectral analysis of this
problem allows us to specify the best sampling strategy in electromagnetic
frequency and predicts a spatially anisotropic distribution of posterior
errors. The reconstruction method is illustrated for reference fields extracted
from realistic magneto-hydrodynamical simulations.
