A mixed norm optimization technique to track the impulse response of the underwater acoustic channel is presented. The channel is assumed to follow a delay-Doppler spread function model, which is known to be sparsely distributed over the complex channel coefficients. The ill-conditioned nature of this estimation problem, along with the need to optimize over a complex field, poses challenges to employing popular sparse estimation techniques that currently exist in the literature. The time-varying nature of the underwater channel also makes efficient tracking difficult. The motivation behind our proposed algorithm is to robustly estimate and efficiently track the sparse delay-Doppler coefficients directly over the complex field. We have formulated the channel estimation problem as that of minimizing over a convex combination of the L1 norm of the sparse coefficients to be estimated and the L2 norm of the estimation error. We analyze the difference between our proposed algorithm to existing L2-constrained L1 optimization techniques and present results showing performance on simulation generated and field data. [Work supported in part by ONR Grant Nos. N00014-05-10085 and N00014-07-10184 and in part by Postdoctoral Scholar Program at the Woods Hole Oceanographic Institution, with funding provided by the Doherty Foundation.]