A system for measuring an acoustic property of material penetrated by a well borehole. The system uses a borehole logging tool comprising preferably one acoustic transmitter and preferably a plurality of acoustic receivers. An independent measurement is used to determine azimuthal orientation of the tool within the borehole. Firings of the transmitter are based upon azimuthal orientation of the tool. Receiver responses resulting from preferably a plurality of transmitter firings per tool revolution are combined to form at least one composite transmitter response. The composite receiver response is compensated for adverse effects of the tool operating eccentered within the borehole.