Summary One major problem with 3D PSDM proccssing resides in the traveltime computation step. Not only this travel time have to be accurate, but the speed of computation has to be high otherwise in certain cases there is a bottleneck in the overall processing turn, around time. The algorithm proposed here for solving the 3D eikonal equation overcomes these two difficulties. A high speed of computation is obtained by using a coarse sampling, but the accuracy of the solutions is ensured by parametrizing the differential equation, not in time, but in celerity. The computations are made on a Cartesian system of coordinates, but the wave-front curvatures are nevertheless implieitly taken into account. Usually, eikonal solutions represent the first arrival traveltimes. In the scheme presented here, the upgoing head-waves are naturally cancelled, as this method proceeds by extrapolating the traveltimes downward. As an option, the undesired downgoing refracted waves can be cancelled by rejecting those wave-fronts that tend to curve towards the vertical axis at the horizontal source location. This approach is efficient for some common geological features.