  Direct seismic imaging of sub-surface flow, sound-speed and magnetic field is
crucial for predicting flux tube emergence on the solar surface, an important
ingredient for space weather. The sensitivity of helioseismic mode-amplitude
cross-correlation to $p$- and $f$-mode oscillations enable formal inversion of
such sub-photospheric perturbations. It is well-known that such problems are
written in the form of an integral equation that connects the perturbations to
the observations via ``sensitivity kernels". While the sensitivity kernels for
flow and sound-speed have been known for decades and have been used
extensively, formulating kernels for general magnetic perturbations had been
elusive. A recent study proposed sensitivity kernels for Lorentz-stresses
corresponding to global magnetic fields of general geometry. The present study
is devoted to proposing kernels for inferring Lorentz-stresses as well as the
solenoidal magnetic field in a local patch on the Sun via Cartesian
mode-coupling. Moreover, for the first time in solar physics, Slepian functions
are employed to parameterize perturbations in the horizontal dimension. This is
shown to increase the number of data constraints in the inverse problem,
implying an increase in the precision of inferred parameters. This paves the
path to reliably imaging sub-surface solar magnetic features in, e.g.,
supergranules, sunspots and (emerging) active regions.
