The ability of pulse-echo measurements to resolve closely spaced reflectors is limited by the duration of the ultrasonic pulse. Resolution can be improved by deconvolution, but this often fails because frequency selective attenuation introduces unknown changes in the pulse shape. In this paper we propose a maximum a posteriori algorithm for simultaneous estimation of a time varying pulse and high-resolution deconvolution. A priori information is introduced to encourage estimates where the pulse varies only slowly and the reflectivity sequence is sparse. This adds sufficient regularization to the problem, and no further assumptions on the pulse such as minimum phase or a particular parametric form are needed. The joint pulse and reflectivity estimate are computed iteratively by alternating steps of pulse estimation and reflectivity estimation. The first step amounts to only a linear least squares fit. The second step is a difficult combinatorial optimization problem that we solve by a suboptimal but efficient search procedure. Due to the sparseness assumption, our approach is particularly suited for layered media containing a limited number of abrupt impedance changes. This is a situation of interest in many applications of nondestructive evaluation. Synthetic and real data results show that the algorithm works well.