Abstract
The timing and amplitudes of arrivals recorded in seismic traces are influenced by velocity variations all along the associated raypaths. Consequently, velocity errors far from the target can lead to blurred imaging of the target body. To partly remedy this problem, we comprehensively reviewed inverting differential traveltimes that satisfied the closure-phase condition. The result is that the source and receiver statics are completely eliminated in the data and velocities far from the target do not need to be known. We successfully used the phase closure equation for traveltime tomography, refraction statics, migration, refraction tomography, and earthquake location, all of which demonstrated the higher resolution achievable by processing data with differential traveltimes rather than absolute traveltimes. More generally, the stationary version of the closure-phase equation is equivalent to Fermat's principle and can be derived from the equations of seismic interferometry. In summary, the general closure-phase equation is the mathematical foundation for approximately redatuming sources and/or receivers to the target of interest without the need to accurately know the statics or the velocity model away from the target.