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We study the elastic wave speed structure of the crust and the uppermost mantle in western Tibet using P wave and S wave arrival times from regional earthquakes recorded by a temporary seismic network. We relocate the earthquakes, and subsequently invert travel time residuals for 3-D distributions of wave speed. Resolution tests with a variety of input structures are used to verify the reliability of our results. The crust beneath western Tibet has low P wave speed (5.9-6.3 km/s) throughout its nearly 80 km thickness, with lower values in this range concentrated within the Lhasa block. Beneath the Himalaya wave speeds are higher. Southern and western limits of the slow material beneath the Tibetan Plateau correlate with the Karakoram fault, and dip beneath the plateau at 40 angle. We find no evidence of a subhorizontal low velocity zone in the crust. In the uppermost mantle, we find a long and narrow region of fast (up to 8.4 km/s) P wave speed extending from the Karakoram fault in NE direction, and crossing the Bangong-Nujiang suture. In a north-south cross section, the distribution of relatively fast P wave speed suggests a ramp-flat geometry consistent with India underthrusting the Tibetan Plateau at least as far as 32.5 N. A plausible interpretation of the upper mantle fast feature is the formation of eclogite from the mafic lower-crustal material of India after it is underthrust beneath Tibet. Notably, in western Tibet this process only takes place in a narrow region.