In the course of a tremendous outburst flood (jökulhlaup) following the subglacial cruption in Vatnajökull, Iceland, in October 1996, a depression in the surface of the ice cap was created as a result of ice melting from the walls of a subglacial tunnel. The surface depression was initially approximately 6 km long, 800 m wide and 100 m deep. This "canyon" represents a significant perturbation in the geometry of the ice cap in this area where the total ice thickness is about 200-400 m. We present results of repeated measurements of flow velocities and elevation changes in the vicinity of the canyon made over a period of about 2 years. The measurements show a reduction in the depth of the canyon and a concomitant decrease in surface flow towards it over time. By calculating the transient evolution of idealized surface depressions using both analytical zeroth- and first-order theories, as well as the shallow-ice approximation (SIA) and a finite-element model incorporating all the terms of the momentum equations, we demonstrate the importance of horizontal stress gradients at the spatial scale of this canyon. The transient evolution of the canyon is calculated with a two-dimensional time-dependent finite-element model with flow parameters (the parameters A and n of Glen's flow law) that are tuned towards an optimal agreement with measured flow velocities. Although differences between measured and calculated velocities are comparable to measurement erros the differences are not randomly distributed. The model is therefore not verified in detail. Nevertheless the model reproduces observed changes in the geometry over a 15 month time period reasonably well. The model also reproduces changes in both velocities and geometry considerably better than an alternative model based on the SIA.