The treatment in the above papers

concerning the effects of a nonvanishing velocity gradient on the waveform and statistical properties of a scattered field and the resulting received reverberation return is modified to include possible focusing phenomena. These, in turn, can become quite pronounced in the neighborhood of caustics and foci in various practical situations vis-á-vis the common unfocused or weakly focused cases.

The modification of earlier results is achieved by inserting an appropriate "focusing factor"

along with the regular absorption factor

for each leg in the propagation path: source

scatter

receiver. Extensions to the "singular" cases, treated more recently by the so-called "modified ray theory

, can also be carried out.