We constrain seismic velocity structure in the Earth's outer core by analyzing differential travel times, waveforms, and amplitude ratios of several pairs of core phases. We use differential travel times and waveforms of PKPbc-PKIKP at 144¿--148¿, PKiKP-PKIKP at 120¿--141¿, PKiKP-PKPBdiff at the PKP caustics distance range (141¿--145¿), and differential travel times and amplitude ratios of PKPab-PKPbc at 146¿--159¿. To avoid the influence of the inner core anisotropy in velocity, we only use the PKiKP-PKIKP and the PKPbc-PKIKP observations whose PKIKP phases sample the inner core along the equatorial paths (paths with their ray angles being larger than 35¿ from the Earth's rotation axis). These observations show the following characteristics: (1) both the observed PKPbc-PKIKP and PKiKP-PKIKP differential travel times show a distinct east-west hemispheric pattern. PKIKP phases arrive about 0.7 s earlier for those sampling the eastern hemisphere (40¿E--180¿E) than those sampling the western hemisphere (180¿W--40¿E); (2) the observed differential PKiKP-PKPBdiff travel time residuals also exhibit a hemispheric difference. PKiKP-PKPBdiff differential travel times are about 0.9 s larger for those sampling the western hemisphere than those sampling the eastern hemisphere; and (3) both the observed PKPab-PKPbc differential travel times and PKPbc/PKPab amplitude ratios show scatter. Overall, these observations can be best explained by two one-dimensional P velocity models, one for each hemisphere, at the bottom of the outer core. The seismic data sampling the eastern hemisphere can be explained by PREM at the bottom of the outer core, while those sampling the western hemisphere can be explained by a lower velocity gradient at the bottom of the outer core, which has reduced velocities relative to PREM linearly increasing from 0% at 200 km above the inner core boundary (ICB) to -0.35% at the ICB. Different velocity gradients at the bottom of the outer core indicate that there may exist a compositional difference and/or a large-scale temperature difference there and that inner core formation processes may be different between the two hemispheres. Different inner core formation processes may produce different geometric inclusions of melt in the top of the inner core, and thus may provide an explanation to the hemispheric variation of seismic velocity and attenuation in the top of the inner core. The seismic data also suggest that the variation of seismic velocity in the tangential cylinder of the outer core, if it exists, is less than 0.1%. We use a compressional wave tomographic model to study the travel time delays caused by the seismic heterogeneities in the mantle. The tomographic model is unable to explain the scatter of the travel time data, suggesting the existence of unknown strong small-scale seismic heterogeneities in the mantle.