Field observations and geodetic data indicate that dike intrusion in volcanic rift zones typically generates normal faulting and graben subsidence at the Earth's surface. Elastic models indicate that two-dimensional (infinite strike length) dikes do not lower the ground surface above the dike and that normal faults do not lower the surface significantly, more than one down-dip fault length from the fault trace. Dikes of finite length produce subsidence above the dike, but not by an appreciable amount, for appropriate dike lengths. Therefore the observed graben subsidence can be achieved only if fault slip extends virtually to the dike plane at depth. Dike intrusion increases the horizontal compression adjacent to the dike and decreases the compression beyond the dike perimeter. Therefore fault slip extending to the dike plane is most likely to occur above or in front of the laterally propagating dike. Two data sets documenting the change in surface elevation accompanying dike intrusion in the Krafla rift zone, Iceland, were inverted to determine the dike and fault geometry at depth. Ten kilometers south of the Krafla caldera, subsidence of a graben 1.5 km wide was produced by fault slip to 1.5-2 km depth, essentially to the dike top. Forty kilometers north of the caldera, subsidence of a graben 6 km wide was produced by fault slip to 4-5 km depth, well within the zone of compression adjacent to the dike. In order to determine if fault slip in front of the dike could have produced the observed subsidence north of the caldera, a three-dimensional boundary element model that computes fault slip during lateral dike propagation was developed. Results indicate that fault slip in front of the dike is capable of producing most of the subsidence observed. Additional subsidence could result from reasonable mechanical anisotropy of the rift zone. The lack of deep fault slip south of the caldera is attributed to a less favorable initial stress state. This is consistent with differences in the tectonic history of the two regions over the past several centuries.