The heavily-cratered southern hemisphere of Mars encompasses the planet's strongest, most widespread magnetization. Our study concentrates on this magnetized region in the southern hemisphere within 40ยฐo f latitude 40ยฐS, longitude 180ยฐW. First we rotate the coordinates to position the center at ร40ยฐ, 180ยฐand treat these new latitudes and longitudes as if they were Cartesian coordinates. Then, using an ordinary two-dimensional Fourier analysis for downward continuation, the MGS (MAG/ER) magnetic field data at satellite mapping elevation of $400 km are extrapolated to 100 km, sources are estimated and used to model the fields. Quantitative comparison of the downward continued field with the aerobraking field for bins having angular deviation within ยฑ30ยฐgives correlation of .947, .868, and .769 for the B r ; B h and B U components, respectively. This agreement of the fields may result from most of the power in the magnetization resting in wavelengths $400 km, with comparatively little at $100 km. Over this region, covering nearly an octant of the planet, just a dozen sources can account for 94% of the variance of the magnetic field at the surface. In these models for the field an obvious asymmetry in polarity exists, with majority of the sources being positive. The locations of strongest surface magnetization appear to be near -but not actually within -ancient multi-ringed basins. We test the likelihood of this association by comparing the observed sources found within and near basins for two alternative basin location scenarios with random distributions. For both alternatives we find the observed distributions to be low-probability occurrences. If contemporaneous, this would establish that Mars' magnetic field extended to the time of impacts causing these basins.