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In ultrasonic imaging systems, the instantaneous pressure at the transducer face during echo reception is typically comprised of many superimposed reflections of a pulse resembling an amplitude-modulated sinusoid. Generally, these reflections are randomly shifted in phase and randomly scaled in amplitude. Moreover, each reflected pulse may have been distorted by passage through a nonuniform medium. The first-order amplitude statistics of such a waveform have long been considered of interest. The backscatter formation process has often been modeled as a random walk in two dimensions. For simplicity, the effects of amplitude-phase dependence and scatterer size distribution have not been fully included in previous work. In most cases of interest this is physically justified; but, given a strongly non-Rayleigh random medium, experience has shown that more accurate expressions may be required. This paper points to the essentials of such an improved analysis. The effects of pulse structure and scatterer size distribution on the statistical properties of the individual step of the random walk are considered de novo. Simulation results are described.