A seismic refraction survey was conducted at Reydarfj¿rdur, Iceland as part of the Iceland Research Drilling Project in July 1978. An array of 15 seismometers was used to measure apparent velocities from 25 explosions along a 23 km refraction line. The experimental yielded data used to determine the P- and S-wave velocity structures beneath the drill site and to obtain bounds on the depth to seismic 'layer 3'. The observed apparent velocities vary continuously as a function of range with a marked increase in velocity at a range of 16 km. This corresponds to an observed cross-over in the travel time data. The measurement of apparent velocities has yielded data of the form X(p), the distance to a measured ray parameter, and &tgr;(p), the delay time. Joint inversion of the X(p) and &tgr;(p) data has been performed by linear programming. Layers of constant slowness gradient are used as the basis for the inversion process. Estimates of error in the data are used to produce extremal bounds on the family of earth models that satisfy the data. When low velocity zones are assumed to be absent, the resulting envelope of P-wave velocity models indicates a continuous variation of velocity with depth. Two regions of high velocity gradient are observed, one near the surface and the other at depths between 2.6 and 3.9 km associated with layer 3 in Iceland. These results suggest that modeling of Iceland's upper crust with a few thick homogeneous constant velocity layers is an inadequate representation of the velocity structure. Some features observed in the seismic data suggest the presence of a low velocity zone with a lid between 1.6 and 2.5 km depth. Rough synnthetic modeling places this lid at the shallower end of this range.