Sparse sensing approaches have been applied to track the delay-Doppler spread in shallow water acoustic communications with varying degrees of success. Tracking the complex-valued delay-Doppler spread function coefficients is challenging due to the ill-conditioned, nonanalytic, and time-varying nature of the channel estimation problem. Since the coefficients typically follow a sparse distribution, fast adaptive least squared approaches fall short of the performance goals. We recently proposed an adaptive sparseoptimization technique, which efficiently tracks the time-varying coefficients by imposing a smooth nonconvex cost function over the L1 norm of the sparse coefficients and the L2 norm of the estimation error. However, a key limitation of this approach is the assumption that the sparsity level of the distribution stays fixed and is uniform across the different Doppler frequencies. Due to uncertainty principles governing time and frequency resolution of nonstationary processes, using uniform averaging window length across the entire Doppler spread also leads to nonuniform resolution of the estimates across the different frequency ranges. We present a multiscale algorithm where we employ variable averaging window length as well as variable sparsity factor to track the delay-Doppler spread across different ranges of frequencies over experimental field data.