Mars Global Surveyor (MGS) measured the most strongly magnetized crust in the heavily cratered southern hemisphere of Mars. Our analysis concentrates on the magnetic lineations or patterns centered near latitude 40 S, longitude 180 W, with a range of values ±40 , using a rotated Cartesian coordinate system. We downward continued the magnetic field measured at 400 km elevation and very closely match the corresponding component measured during the aerobraking phase at altitudes extending down to 100 km. Using the vertical component of the magnetic field alone, we construct a unique scalar potential and independently obtain from the derivatives of this scalar potential the x and y components of the field. These derived components agree very well with the observed horizontal components. This demonstrates the validity and utility of the method and the Cartesian approximation, and also it confirms the consistency of the MGS magnetic data set. A model constructed with just 8 vertical dipoles accounts for 80% of the variance of the scalar potential at 400 km over the region analyzed, but 14 dipoles can account for only 64% of the variance at 100 km. We also construct the vector potential, the curl of which generates the three components of the magnetic field. This more complicated description may contain more physical meaning than the scalar potential. The vector potential shows abrupt changes in direction over the analyzed region, suggesting either different stages of magnetization or local demagnetization.