Downward-looking Linear Array Synthetic Aperture Radar (LASAR) has many potential applications in the topographic mapping, disaster monitoring and reconnaissance applications, especially in the mountainous area. However, limited by the sizes of platforms, its resolution in the linear array direction is always far lower than those in the range and azimuth directions. This disadvantage leads to the blurring of Three-Dimensional (3D) images in the linear array direction, and restricts the application of LASAR. To date, the research on 3D SAR image enhancement has focused on the sparse recovery technique. In this case, the one-to-one mapping of Digital Elevation Model (DEM) brakes down. To overcome this, an optimal DEM reconstruction method for LASAR based on the variational model is discussed in an effort to optimize the DEM and the associated scattering coefficient map, and to minimize the Mean Square Error (MSE). Using simulation experiments, it is found that the variational model is more suitable for DEM enhancement applications to all kinds of terrains compared with the Orthogonal Matching Pursuit (OMP)and Least Absolute Shrinkage and Selection Operator (LASSO) methods.